Limits and Derivatives

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New answer posted•a year ago

Limits and Derivatives Class 11th

The sum of the roots of the equation, $x + 1 - 2 l o g_{2} (3 + 2^{x}) + 2 l o g_{4} (10 - 2^{- x}) = 0,$ is :

3 Views|Posted a year ago

Asked by Vishal Baghel

Answered by

Vishal Baghel | Contributor-Level 10

a year ago

$x + 1 - 2 l o g_{2} (3 + 2^{x}) + 2 l o g_{4} (10 - 2^{- x}) = 0$

$x + 1 + l o g_{2} [\frac{10 . 2^{x} - 1}{{(3 + 2^{x})}^{2}}] - x = 0$

${(2^{x})}^{2} - 14 . 2^{x} + 11 = 0$

2x + y

$x_{1} + x_{2} = l o g_{2} (49 - \frac{152}{4})$

$x_{1} + x_{2} = l o g_{2} 11$

Upvote

New answer posted•a year ago

Limits and Derivatives Ncert Solutions Maths class 11th

42. Find the derivative of the following functions:

(i)sin x cos x (ii) $s e c x$ (iii)5 sec x + 4 cosx

(iv) $c o s e c x$ (v)3cot x + 5 cosec x

(vi) $5 s i n x - 6 c o s x + 7$ (vii) $2 t a n x - 7 s e c x$

12 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

(i) f(x)=sin x cos x

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{s i n (x + h) c o s (x + h) ? s i n x c o s x}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} \times [2 s i n (x + h) c o s (x + h) ? 2 s i n x c o s x]$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} [s i n 2 (x + h) ? s i n 2 x]$

$= \underset{h ? 0}{l i m} \frac{1}{2 h} [2 c o s \frac{2 (x + h) + 2 x}{2} s i n \frac{2 (x + h) ? 2 x}{2}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [c o s (2 x + h) s i n h]$

$= \underset{h ? 0}{l i m} c o s (2 x + h) \times \underset{h ? 0}{l i m} \frac{s i n h}{h}$

$= c o s (2 x + 0)$

$= c o s 2 x$

$(ii) f (x) = s e c x$

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{h} [s e c (x + h) ? s e c x]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{1}{c o s (x + h)} ? \frac{1}{c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{c o s x ? c o s (x + h)}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{? 2 s i n (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [? \frac{2 s i n (\frac{2 x + h}{2}) s i n (? h / 2)}{c o s (x + h) c o s x}]$

$= \underset{h ? 0}{l i m} \frac{(? 1 s i n (\frac{2 x + h}{2})}{c o s (x + h) c o s x} \times \underset{h ? 0}{l i m} (? 1) \frac{s i n h / 2}{h / 2}$

$= \frac{s i n x}{c o s x ? c o s x} \times 1$

$= t a n x ? s e c x .$

(iii) Given f(x)=5 sec x+4 cosx.

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$\begin{array}{l} = \underset{h ? 0}{l i m} \frac{5 s e c (x + h) + 4 c o s (x + h) ? [5 s e c x + 4 c o s x]}{h} . \end{array}$

$= \underset{h ? 0}{l i m} \frac{5}{h} [s e c (x + h) ? s e c x] + \underset{h ? 0}{l i m} \frac{4}{h} [c o s (x + h) ? c o s x]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [\frac{1}{c o s (x + h)} ? \frac{1}{c o s x}] + \underset{h ? 0}{l i m} \frac{4}{h} [? 2 s i n (\frac{x + h + x}{2}) s i n (\frac{x + h ? x}{2})]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [\frac{c o s x ? c o s (x + h)}{c o s (x + h) (c o s x)}] + \underset{h ? 0}{l i m} \frac{4}{h} [? 2 s i n (\frac{2 x + h}{2}) s i n \frac{h}{2}]$

$= \underset{h ? 0}{l i m} \frac{5}{h} [? \frac{2 s i n (\frac{2 x + h}{2}) s i n (? h / 2)}{c o s (x + h) c o s x}] ? 4 \underset{h ? 0}{l i m} s i n (\frac{2 x + h}{2}) \underset{h ? 0}{l i m} \frac{s i n h / 2}{h / 2}$

$= \frac{s i n (\frac{2 x + 0}{2})}{c o s (x + 0) c o s x} \times 1 ? 4 s i n (\frac{2 x}{2})$

$= 5 \frac{s i n x}{c o s x} ? \frac{1}{c o s x} ? 4 s i n x$

$= 5 t a n x ? s e c x ? 4 s i n x$

$(iv) Given f (x) = c o s e c x$

$f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$

$= \underset{h ? 0}{l i m} \frac{1}{h} [c o s e c (x + h) ? c o s e c x]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{1}{s i n (x + h)} ? \frac{1}{s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{s i n x ? s i n (x + h)}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})]}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{1}{h} [\frac{2 c o s (\frac{2 x + h}{2}) s i n (? h / 2)]}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{c o s (\frac{2 x + h}{2})}{s i n (x + h) s i n x} \times \frac{(? 1) s i n (2)}{h / 2})$

$= \frac{c o s (\frac{2 x + 0}{2})}{s i n (x + 0) s i n x} \times (? 1)$

$= ? \frac{c o s x}{s i n x} \times \frac{1}{s i n x}$

$= ? c o t x ? c o s e c x$

(v) Given,f(x)=3 cot x+5cosecx.

So, $f^{?} (x) = \underset{h ? 0}{l i m} \frac{f (x + h) ? f (x)}{h}$ $= \frac{2}{c o s (x + 0) c o s x} \times 1 + 7 s i n (\frac{2 x + 0}{2}) \times (? 1)$

$= \underset{h ? 0}{l i m} 3 c o t (x + h) + 5 c o s e c (x + h) ? [3 c o t x + 5 c o s x)$

$_{h ? 0} \frac{3}{h} [c o t (x + h) ? c o t x] + \underset{h ? 0}{l i m}$

$= ? \frac{5}{h} [c o s e c (x + h) ? c o s e c x]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{c o s (x + h)}{s i n (x + h)} ? \frac{c o s x}{s i n x}] + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{1}{s i n (x + h)} ? \frac{1}{s i n x}]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{c o s (x + h) s i n x ? c o s x s i n (x + h)}{s i n (x + h) s i n x}] + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{s i n x ? s i n (x + h)}{s i n (x + h) s i n x}]$

$= \underset{h ? 0}{l i m} \frac{3}{h} [\frac{s i n (x ? (x + h))}{s i n (x + h) s i n x} + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{2 c o s (\frac{x + x + h}{2}) s i n (\frac{x ? (x + h)}{2})}{s i n (x + h) s i n x}$

$= \underset{h ? 0}{l i m} \frac{3}{s i n (x + h) s i n x} \times \underset{h ? 0}{l i m} (? 1) \frac{s i n h}{h} + \underset{h ? 0}{l i m} \frac{5}{h} [\frac{2 c o s (\frac{2 x + h}{2}) s i n (? h / 2)}{s i n (x + h) s i n x}$

$= \frac{3}{s i n x ? s i n x} \times (? 1) + 5 \underset{h...Read moreUpvoteNew answer posted • a year agoLimits and Derivatives Ncert Solutions Maths class 11th 29. Kindly consider the following 4 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 29. Given, f (x) = (x - a 1) (x - a 2)… (x - a n)So, \underset{x \to a_{1}}{l i m} f (x) = \underset{x \to a_{1}}{l i m} (x - a_{1}) \underset{x \to a_{1}}{l i m} (x - a_{2}) ? \underset{x \to a_{1}}{l i m} (x - a^{n}) = (a 1 - a 1) (a 1 - a 2) … (a 1 - a n) = 0 (a 1 - a 2) … (a 1 - a n) = 0 And \underset{x \to a}{l i m} f (x) = \underset{x \to a}{l i m} (x - a_{1}) \underset{x \to a}{l i m} (x - a_{2}) \dots \underset{x \to a}{l i m} (x - a n) = (a - a 1) (a - a 2) … (a - a n) UpvoteNew answer posted • a year agoLimits and Derivatives Ncert Solutions Maths class 12th 17. Kindly consider the following 6 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 17. Kindly go through the solution = = \underset{x \to 0}{l i m} \frac{2 s i n^{2} x}{2 s i n^{2} \frac{x}{2}} = \frac{\underset{x \to 0}{l i m} {(\frac{s i n x}{x})}^{2} \times x^{2}}{\underset{x \to 0}{l i m} {(\frac{s i n^{} \frac{x}{2}}{\frac{x}{2}})}^{2} {\frac{x}{2}}^{2}} = \frac{{(1)}^{2} \times x^{2} \times 4}{{(1)}^{2} \times x^{2}} = 4 UpvoteNew answer posted • a year agoLimits and Derivatives Limits and Derivatives 18. \underset{x \to 0}{l i m} \frac{a x + x c o s x}{b s i n x} 6 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 18. \underset{x ? 0}{l i m} \frac{a x + x c o s x}{b s i n x} = \underset{x ? 0}{l i m} \frac{x (a + c o s x)}{b s i n x} = \frac{1}{b} * \frac{(a + c o s 0)}{1} = \frac{a + 1}{b} Upvote View All 2 AnswersNew answer posted • a year agoLimits and Derivatives Limits and Derivatives Kindly consider the following 73. \frac{x}{s i n^{n} x} 5 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 73. Given, f (x) = \frac{x}{s i n^{n} x} So, f?(x) = \frac{(s i n^{x} x \frac{d x}{d x}) - (x \frac{d}{d x} s i n^{x} x)}{{(s i n^{x} x)}^{2}} = \frac{s i n^{x} x - x \cdot n {(s i n x)}^{x - 1} \frac{d}{d x} s i n x}{{(s i n x)}^{2 x}} = \frac{{(s i n x)}^{n} - x \cdot x {(s i n x)}^{n - 1} \cdot c o s x}{{(s i n x)}^{2 n}} = \frac{{(s i n x)}^{n - 1} [s i n x - n x \cdot c o s x]}{{(s i n x)}^{2 n}} = \frac{s i n x - n \cdot x \cdot c o s x}{{(s i n x)}^{2 n - (n - 1)}} = \frac{s i n x - n x c o s x}{{(s i n x)}^{n + 1}} UpvoteNew answer posted • a year agoLimits and Derivatives Limits and Derivatives Kindly consider the following 72. (x + sec x) (x - tan x 5 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 72. Given, f (x) = (x + sec x) (x tan x) So, f?(x) = (x + see x) \frac{d}{d x} (x - t a n x) + (x - t a n x) \frac{d}{d x} (x + s e c x) . = (x + s e c x) (\frac{d x}{d x} - \frac{d}{d x} t a n x) + (x - t a n x) (\frac{d x}{d x} + \frac{d}{d x} s e c x) Let g(x) = tan x. So, g?(x) = = \underset{h \to 0}{l i m} \frac{1}{h} [t a n (x + h) - t a n x] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h)}{c o s (x + h)} - \frac{s i n x}{c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h) c o s x - s i n x c o s (x + h)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{s i n (x + h - x)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{c o s (x + h) c o s x} \times \underset{h \to 0}{l i m} \frac{s i n h}{h} = \frac{1}{c o s^{2} x} = s e c^{2} x . _____ (2) And P(x) = see x. = \underset{h \to 0}{l i m} \frac{s e c (x + h) - s e c x}{h} = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{1}{c o s (x + h)} - \frac{1}{c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{c o s x - c o s (x + h)}{c o s (x + h) c o s x}] = \underset{h \to 0}{l i m} \frac{1}{h} [\frac{- 2 s i n (\frac{x + x + h}{2}) s i n (\frac{x - (x + h)}{2})}{c o s (x + h) c o s x}] = \frac{s i n x}{c o s^{2} x} = tan x sec x. ____ (3) Putting (2) and (3) in (1) we get, f ?(x) = (x + sec x) (1 - sec 2 x) + (x - tan x) (1 + tan x ...Read more UpvoteNew answer posted • a year agoLimits and Derivatives Limits and Derivatives Kindly consider the following 71. \frac{x}{1 + t a n x} 8 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 71. Given, f (x) = \frac{x}{1 + t a n x} = \frac{x}{1 + \frac{s i n x}{c o s x}} = \frac{c o s x \cdot x}{c o s x + s i n x} S o, f^{'} (x) = \frac{(c o s x + s i n x) \frac{d}{d x} (x \cdot c o s x) - (x \cdot c o s x) \frac{d}{d x} (c o s x + s i n x)}{{(c o s x + s i n x)}^{2}} = \frac{(c o s x + s i n x) (- x s i n x + c o s x) - (- x \cdot s i n x c o s x + x c o s^{2} x)}{{(c o s x + s i n x)}^{2}} = \frac{- x c o s x s i n x + c o s^{2} x - x s i n^{2} x + s i n x c o s x + x s i n x c o s x - x c o s^{2} x}{{(c o s x + s i n x)}^{2}} = \frac{c o s^{2} x + s i n x c o s x - x (s i n^{2} x + c o s^{2} x)}{{(c o s x + c s i n x)}^{2}} Dividing numerator and denominator by cos2x we get, = \frac{\frac{c o s^{2} x}{c o s^{2} x} + \frac{s i n x}{c o s x} \cdot \frac{c o s x}{c o s x} - x \frac{1}{c o s^{2} x}}{{(\frac{c o s x}{c o s x} + \frac{s i n x}{c o s x})}^{2}} = \frac{1 + t a n x - x s e c^{2} x}{{(1 + t a n x)}^{2}} UpvoteNew answer posted • a year agoLimits and Derivatives Limits and Derivatives Kindly consider the following 70. \frac{x^{2} c o s (\frac{π}{4})}{s i n x} 2 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 70. Given, f (x) = \frac{x^{2} c o s (\frac{π}{4})}{s i n x} = \frac{x^{2}}{s i n x} \times c o s \frac{π}{4} So, f? (x) = = c o s \frac{π}{4} [\frac{2 \cdot x s i n x - x^{2} c o s x}{s i n^{2} x}] = x c o s \frac{π}{4} [\frac{2 s i n x - x c o s x}{s i n^{2} x}] . UpvoteNew answer posted • a year agoLimits and Derivatives Limits and Derivatives Kindly consider the following 69. \frac{4 x + 5 s i n x}{3 x + 7 c o s x} 8 Views | Posted a year ago Asked by Alok Kumar Singh Share Follow Answer S Answered by Alok Kumar Singh | Contributor-Level 10 a year ago 69. Given, f (x) = \frac{4 x + 5 s i n x}{3 x + 7 c o s x} So, f^{'} (x) = \frac{(3 x + 7 c o s x) \frac{d}{d x} (4 x + 5 s i n x) - (4 x + 5 s i n x) \frac{d}{d x} (3 x + 7 c o s x)}{{(3 x + 7 c o s x)}^{2}} = \frac{(3 x + 7 c o s x) (4 x + 5 c o s x) - (4 x + 5 s i n x) (3 - 7 s i n x)}{{(3 x + 7 c o s x)}^{2}} \begin{array}{l} \underline{= x + x x + x +^{} x - x + x x - x + 35 s i n^{2} x} \\ {(3 x + 7 c o s x)}^{2} \end{array} = \frac{15 x c o s x + 28 c o s x + 28 x s i n x - 15 s i n x + 35 (c o s^{2} x + s i n^{2} x)}{{(3 x + 7 c o s x)}^{2}} = \frac{35 + 15 x c o s x + 28 c o s x + 28 x s i n x - 15 s i n x}{{(3 x + 7 c o s x)}^{2}} UpvotePrev 12345678910 NextRegister to get relevant Questions & Discussions on your feed Login Or Register Share Your College Life Experience Write A Review Community Guidelines User Points System Questions Discussions TagsGet our experts to answer your questions within 24 Hrs Write your questions here Ask Question MBA MBA Top MBA Colleges MBA Colleges Executive MBA Colleges MBA Exams CAT MAT Online MBA MBA College Predictors Engineering Engineering Top Engineering Colleges Engineering Colleges Engineering Exams JEE Main JEE Advanced Engineering College Predictors Medicine NEET UG NEET PG NEET SS NEET MDS INI CET FMGE AIAPGET Top Medical Colleges Medical Colleges Medical Exams Other Courses Animation B.Com B.Sc BBA CA Fashion Designing Hotel Management Law Mass Communication & Media MBBS Sarkari Exams RRB Group D RRB NTPC CTET UPTET UGC NET DSSSB SSC CGL SSC CHSL SSC GD NDA Resources Careers after 12th Courses After 12th Ask a Question Write a college review Articles Law College Predictors Hospitality College Predictor NCERT Solutions NCERT Solutions Class 12 Maths NCERT Solutions Class 12 Physics Important Updates CAT Exam CAT Admit Card 2026 NMAT Exam BITS Pilani cmat registration 2027 SNAP Exam CAT Exam Syllabus UP Board Exam Date 2027 XAT Exam CMAT Exam Study Abroad Study Abroad Home BTech abroad MBA abroad MS abroad GRE GMAT SAT IELTS TOEFL Get App, Its faster and better Get on Play Store Get on App Store About Shiksha About Us Management Team Careers Shiksha Authors FAQs Contact Us Our Group Infoedge.in Naukri.com Naukri Resume Maker Naukrigulf 99acres.com Jeevansathi.com AmbitionBox.com Shiksha Abroad Job Hai Enterprise Client Login Advertising/Sales Enquiries Add Colleges Get in Touch 8585951111 Contribute Write Review Follow Us A Group Company Grievances Notices / Summons Privacy Sitemap Terms & Conditions Trade Marks belong to the respective owners. 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\u003cmrow> \u003cmi>g\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmrow> \u003cmo>(\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmn>0\u003c/mn> \u003cmo>−\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>0\u003c/mn> \u003cmo>,\u003c/mo> \u003c/mrow> \u003c/math> \u003c/span>is :\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-the-sum-of-the-roots-of-the-equation-x-1-2-l-o-g-2-3-2-x-2-l-o-g-4-1-0-2-x-0-if-gte-mso-9-xml-xml-qna-11778080","answerCount":1,"viewCount":3,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2929806,"tagName":"Class 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\u003cmo>+\u003c/mo> \u003cmn>2\u003c/mn> \u003cmi>l\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsub> \u003cmrow> \u003cmi>g\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>4\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmn>0\u003c/mn> \u003cmo>−\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmo>−\u003c/mo> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003cmo>=\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmi>x\u003c/mi> \u003cmo>+\u003c/mo> \u003cmn>1\u003c/mn> \u003cmo>+\u003c/mo> \u003cmi>l\u003c/mi> \u003cmi>o\u003c/mi> \u003cmsub> \u003cmrow> \u003cmi>g\u003c/mi> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msub> \u003cmrow> \u003cmo> [\u003c/mo> \u003cmrow> \u003cmfrac> \u003cmrow> \u003cmn>1\u003c/mn> \u003cmn>0\u003c/mn> \u003cmo>.\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/msup> \u003cmo>−\u003c/mo> \u003cmn>1\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmn>3\u003c/mn> \u003cmo>+\u003c/mo> \u003cmsup> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> \u003cmi>x\u003c/mi> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003cmo>)\u003c/mo> \u003c/mrow> \u003c/mrow> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003c/msup> \u003c/mrow> \u003c/mfrac> \u003c/mrow> \u003cmo>]\u003c/mo> \u003c/mrow> \u003cmo>−\u003c/mo> \u003cmi>x\u003c/mi> \u003cmo>=\u003c/mo> \u003cmn>0\u003c/mn> \u003c/mrow> \u003c/math> \u003c/span>\u003c/p>\u003cp>\u003cspan contenteditable=\"false\"> \u003cmath> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmrow> \u003cmo> (\u003c/mo> \u003cmrow> \u003cmsup> \u003cmrow> \u003cmn>2\u003c/mn> \u003c/mrow> \u003cmrow> 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\u003c/span>\u003c/p>","answerOwnerUserId":56416565,"answerOwnerName":"Vishal Baghel","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11586144,"title":"\u003cp>\u003cstrong>42. Find the derivative of the following functions:\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(i)sin x cos x \u003cmath>\u003cmrow>\u003c/mrow>\u003c/math> (ii)\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> (iii)5 sec x + 4 cosx\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003c/mrow>\u003c/math>(iv) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span> (v)3cot x + 5 cosec x\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>(vi) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>5\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>6\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>7\u003c/mn>\u003c/mrow>\u003c/math>\u003c/span> (vii) \u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>7\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-42-find-the-derivative-of-the-following-functions-i-sin-x-cos-x-ii-secx-iii-5-sec-x-4-cosx-iv-qna-11586144","answerCount":1,"viewCount":12,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and 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stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy 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stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo 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stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>?\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>5\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>/\u003c/mo>\u003cmn>2\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>3\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>?\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmo>?\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmn>5\u003c/mn>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11586058,"title":"\u003cp>\u003cstrong>\u003cpicture>29. Kindly consider the following\u003c/picture>\u003c/strong>\u003c/p>\n\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742463703phpBS7vhd.jpeg\" alt=\" width='1200' height=\" width=\"764\" height=\"71\">\u003c/picture>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-29-kindly-consider-the-following-qna-11586058","answerCount":1,"viewCount":4,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2864885,"tagName":"Ncert Solutions Maths class 11th","tagType":"Exams","url":"https://www.shiksha.com/preparation/maths-ncert-solutions-class-11th-ns","type":"Exams"}],"activityTime":"a year ago","creationDate":"a year ago","questionCreationDate":"a year ago","questionId":11586058,"questionOwnerId":null,"questionOwnerName":"alok kumar","threadOwnerUserId":56416739,"threadStatus":"live","isHtml":1,"discussionUserId":null,"discussionTxtId":null,"answerId":11586064,"answerText":"\u003cp>\u003cstrong>29. \u003c/strong>\u003c/p>\u003cp>Given, \u003cem>f\u003c/em> (\u003cem>x\u003c/em>) = (\u003cem>x\u003c/em>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>x\u003c/em>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>)… (\u003cem>x\u003c/em>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp>So, \u003cstrong> \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>? \u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>= (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp>= 0 (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>\u003csub>1\u003c/sub>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>\u003cp> = 0\u003c/p>\u003cp>And\u003cstrong> \u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmi>f\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmrow>\u003cmo> (\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsub>\u003cmrow>\u003cmi>a\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msub>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>…\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmi>a\u003c/mi>\u003c/mrow>\u003c/munder>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>= (\u003cem>a\u003c/em>-\u003cem>a\u003c/em>\u003csub>1\u003c/sub>) (\u003cem>a\u003c/em>-\u003cem>a\u003c/em>\u003csub>2\u003c/sub>) … (\u003cem>a\u003c/em>-\u003cem>a\u003csub>n\u003c/sub>\u003c/em>)\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11586012,"title":"\u003cp>\u003cstrong>17. Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462933phpaZMuEe.jpeg\" alt=\"\" width=\"200\" height=\"62\">\u003c/picture>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-17-kindly-consider-the-following-qna-11586012","answerCount":1,"viewCount":6,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2878357,"tagName":"Ncert Solutions Maths class 12th","tagType":"Exams","url":"https://www.shiksha.com/preparation/maths-ncert-solutions-class-12th-ns","type":"Exams"}],"activityTime":"a year ago","creationDate":"a year ago","questionCreationDate":"a year ago","questionId":11586012,"questionOwnerId":null,"questionOwnerName":"alok kumar","threadOwnerUserId":56416739,"threadStatus":"live","isHtml":1,"discussionUserId":null,"discussionTxtId":null,"answerId":11586018,"answerText":"\u003cp>\u003cstrong>\u003cpicture>17. Kindly go through the solution\u003c/picture>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742462989phpO7uSFs.jpeg\" alt=\"\" width=\"518\" height=\"90\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>=\u003c/math>\u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmfrac bevelled=\"true\">\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo> 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stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmn>4\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>×\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>= 4\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 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mathml\">\u003cmath>\u003cmrow>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-18-limx-0ax-xcosxbsinx-qna-11345776","answerCount":2,"viewCount":6,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2258613,"tagName":"Limits and 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\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>? \u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731061609phpmS3cGH.jpeg\" alt=\"\" width=\"175\" height=\"94\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>*\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmo stretchy=\"false\"> (\u003c/mo>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmn>0\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>a\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>b\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11356192,"title":"\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">Kindly consider the following\u003c/span>\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">73. \u003c/span>\u003c/strong>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-kindly-consider-the-following-73-xsinnx-qna-11356192","answerCount":1,"viewCount":5,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2258613,"tagName":"Limits and 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mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>−\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/p>\u003cp>=\u003cstrong> 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stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>·\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>n\u003c/mi>\u003cmo>−\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>n\u003c/mi>\u003cmo>+\u003c/mo>\u003cmn>1\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11356162,"title":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>72. (\u003cem>x\u003c/em> + sec \u003cem>x\u003c/em>) (\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-kindly-consider-the-following-72-x-sec-x-x-tan-x-qna-11356162","answerCount":1,"viewCount":5,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and Derivatives","tagType":"ExamGroups","url":"https://www.shiksha.com/tags/limits-and-derivatives-tdp-2933282","type":"ne"},{"tagId":2258613,"tagName":"Limits and Derivatives","tagType":"Exams","url":"https://www.shiksha.com/preparation/maths-limits-and-derivatives-tp","type":"Exams"}],"activityTime":"a year ago","creationDate":"a year ago","questionCreationDate":"a year ago","questionId":11356162,"questionOwnerId":null,"questionOwnerName":"alok kumar","threadOwnerUserId":56416739,"threadStatus":"live","isHtml":1,"discussionUserId":null,"discussionTxtId":null,"answerId":11356166,"answerText":"\u003cp>\u003cstrong>72. \u003c/strong>Given, f (x) = (x + sec x) (x tan x)\u003c/p>\u003cp>So, f?(x) = (x + see x)\u003cstrong> \u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmo>+\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>d\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>d\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471139phpRI5F40_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471139phpRI5F40.jpeg\" alt=\"\" width=\"458\" height=\"46\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>Let g(x) = tan x.\u003c/p>\u003cp>So, g?(x)\u003cstrong> = \u003c/strong>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471188phpFPN8R5.jpeg\" alt=\"\" width=\"252\" height=\"42\">\u003c/p>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo stretchy=\"false\">[\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmo>−\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">]\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742470962php23HO2z_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742470962php23HO2z.jpeg\" alt=\"\" width=\"191\" height=\"66\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>=\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/math>\u003c/strong> . _____ (2)\u003c/p>\u003cp>And P(x) = see x.\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471001phpwSwYwX_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471001phpwSwYwX.jpeg\" alt=\"\" width=\"246\" height=\"57\">\u003c/picture>\u003c/div>\u003c/div>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmi>c\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo 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stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmunder>\u003cmrow>\u003cmi>l\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>m\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003cmo>→\u003c/mo>\u003cmn>0\u003c/mn>\u003c/mrow>\u003c/munder>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>h\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo>[\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmo>−\u003c/mo>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>h\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cdiv>\u003cdiv>\u003cpicture>\u003csource srcset=\"https://images.shiksha.com/mediadata/images/articles/1742471046phpjySO16_480x360.jpeg\" media=\"(max-width: 500px)\">\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742471046phpjySO16.jpeg\" alt=\"\" width=\"322\" height=\"271\">\u003c/picture>\u003c/div>\u003c/div>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math> \u003c/strong>= tan x sec x. ____ (3)\u003c/p>\u003cp> Putting (2) and (3) in (1) we get,\u003c/p>\u003cp>\u003cem> f\u003c/em>?(\u003cem>x\u003c/em>) = (\u003cem>x\u003c/em> + sec \u003cem>x\u003c/em>) (1 - sec\u003csup>2\u003c/sup>\u003cem>x\u003c/em>) + (\u003cem>x\u003c/em>- tan \u003cem>x\u003c/em>) (1 + tan \u003cem>x\u003c/em> sec \u003cem>x\u003c/em>)\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11356136,"title":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>71.\u003c/strong> \u003cstrong>\u003cspan title=\"Click to copy mathml\">\u003cmath>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/span>\u003c/strong>\u003c/p>","url":"https://ask.shiksha.com/preparation-maths-kindly-consider-the-following-71-x1-tanx-qna-11356136","answerCount":1,"viewCount":8,"followerCount":0,"tags":[{"tagId":2933282,"tagName":"Limits and 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stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>+\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>Dividing numerator and denominator by cos2x we get,\u003c/p>\u003cp>\u003cpicture>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1731318417phpZfJ4cO.jpeg\" alt=\"\" width=\"195\" height=\"76\">\u003c/picture>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>·\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>s\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmrow>\u003cmo>(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>+\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>e\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>c\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmo stretchy=\"false\">(\u003c/mo>\u003cmn>1\u003c/mn>\u003cmo>+\u003c/mo>\u003cmi>t\u003c/mi>\u003cmi>a\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo stretchy=\"false\">)\u003c/mo>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 10","answerOwnerImage":null,"aboutMe":"","commentCount":0,"likeCount":0,"dislikeCount":0,"hasUserVotedUp":false,"hasUserVotedDown":false,"isUserFollowing":false,"isThreadOwner":false,"hasUserAnswered":false,"suggestedInstituteIds":null},{"type":"Q","uniqueId":"1","id":11356122,"title":"\u003cp>\u003cstrong>Kindly consider the following\u003c/strong>\u003c/p>\n\u003cp>\u003cstrong>70. 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(\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>)\u003c/mo>\u003c/mrow>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmfrac>\u003cmrow>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003cmo>×\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>So, f? (x) \u003cstrong>= \u003c/strong>\u003cimg src=\"https://images.shiksha.com/mediadata/images/articles/1742469923phprlFy9G.jpeg\" alt=\"\" width=\"266\" height=\"120\">\u003c/p>\u003cp> \u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmo>·\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmsup>\u003cmrow>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>\u003cp>\u003cstrong>\u003cmath>\u003cmrow>\u003cmo>=\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmfrac>\u003cmrow>\u003cmi>π\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>4\u003c/mn>\u003c/mrow>\u003c/mfrac>\u003cmrow>\u003cmo> [\u003c/mo>\u003cmrow>\u003cmfrac>\u003cmrow>\u003cmn>2\u003c/mn>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmi>n\u003c/mi>\u003cmi>x\u003c/mi>\u003cmo>−\u003c/mo>\u003cmi>x\u003c/mi>\u003cmi>c\u003c/mi>\u003cmi>o\u003c/mi>\u003cmi>s\u003c/mi>\u003cmi>x\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmi>s\u003c/mi>\u003cmi>i\u003c/mi>\u003cmsup>\u003cmrow>\u003cmi>n\u003c/mi>\u003c/mrow>\u003cmrow>\u003cmn>2\u003c/mn>\u003c/mrow>\u003c/msup>\u003cmi>x\u003c/mi>\u003c/mrow>\u003c/mfrac>\u003c/mrow>\u003cmo>]\u003c/mo>\u003c/mrow>\u003cmo>.\u003c/mo>\u003c/mrow>\u003c/math>\u003c/strong>\u003c/p>","answerOwnerUserId":56416739,"answerOwnerName":"alok kumar singh","answerOwnerLevel":"Contributor-Level 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var imported = document.createElement('script');
//imported.src = 'https://js.shiksha.ws/public/js/chatplugin.js?v=24920259ta';
imported.src = 'https://js.shiksha.ws/public/js/chatplugin.js?v='+d.getDate()+''+(d.getMonth()+1)+''+d.getFullYear()+'22THSep';
// imported.title = 'https://js.shiksha.ws/public/js/build/';
imported.async = 1;

// imported.defer = 1;
document.head.appendChild(imported);
}
}

window.isAbroad = false;

if(window.assistantInit === false || typeof(window.assistantInit) === "undefined"){
(window.isAbroad) ? showShikshaAssistant() : setTimeout(()=>{showShikshaAssistant()},3000);
}
window.hideGdpr = 0;
window.isInternationalUser = true;
window.ENV = 'production';
window.adSenseScriptStatus = 0;
window.fromWherePWA = "TagDetailPageData";
window.isPartytownEnabled = 'no' == 'yes';

// window.showGoogleAdSense = 'no';

const scrollToElementById = (elementId, scrollOffset = 160) => {
const element = document.getElementById(elementId);
if(element){
window.scrollTo({
behavior: "smooth",
top: element.getBoundingClientRect().top - document.body.getBoundingClientRect().top - scrollOffset,
});
}
}

const isTargetElementClicked = (target, targetCategory) => {
switch(targetCategory) {
case 'chat':
return target.closest('.chat--prompt-col, .prompt-chatboat, .primary-chat-icon');
case 'close':
return target.closest('.chat--prompt_close');
case 'exam':
return fromWherePWA != "applyPage" && target.closest('h1');
case 'noclicks':
return target.closest('#college-search-input, #search-with-suggester-input, #filter-search-input, #flticon');
case 'tocAnchor':
return target.closest('.toc-anchor');
default:
return false;
}
}

// Interceptor listener — MUST be in capture phase before gtm
document.addEventListener('click', function (e) {
const target = e.target;

if (isTargetElementClicked(target, 'chat')) {
console.log('toggle chat');
e.stopImmediatePropagation();
setTimeout(toggleChatWindow, 50);
}
else if (isTargetElementClicked(target, 'close')) {
console.log('close chat');
e.stopImmediatePropagation();
setTimeout(() => {
closeChatPrompt();
showChatNotification();
}, 50);
}
else if(isTargetElementClicked(target, 'exam')){
e.stopImmediatePropagation()
}
else if(isTargetElementClicked(target, 'noclicks')){
e.stopImmediatePropagation();
}
else if(isTargetElementClicked(target, 'tocAnchor')){
e.stopImmediatePropagation();
e.preventDefault();
setTimeout(() => {
const liId = target.parentElement?.id || "li_" + target.href?.split("#")?.[1];
const liElement = document.getElementById(liId);
if(liElement){
liElement.click();
}
}, 50);
}
}, true); // capture = true

// Optimized GTM initialization for better INP score
let gtmInitialized = false;
let gtmInitScheduled = false;

// Modern scheduling utilities
function scheduleGTMInit() {
if (gtmInitScheduled) return;
gtmInitScheduled = true;

// Use scheduler API if available, fallback to requestIdleCallback
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
loadGTMScripts();
}, { priority: 'background' });
} else if ('requestIdleCallback' in window) {
requestIdleCallback(() => {
loadGTMScripts();
}, { timeout: 2000 });
} else {
// Fallback for older browsers
setTimeout(() => {
loadGTMScripts();
}, 10000);
}
}
//Important: moving this to App.js for experiment
if('no' == 'yes'){
setTimeout(function(){lazyLoadJs(document,"script", ["https://www.googletagmanager.com/gtag/js?id=G-K11YYW39J9"],true);
window.dispatchEvent(new CustomEvent('ptupdate'));},0)
}

function loadGTMScripts() {
if (gtmInitialized) return;
gtmInitialized = true;

// Remove event listeners immediately to prevent re-triggering
document.body.removeEventListener('click', handleUserInteraction);
window.removeEventListener('scroll', handleUserInteraction, { passive: true });

// Load GTM with proper scheduling
if ('scheduler' in window && 'postTask' in window.scheduler) {
setTimeout(() => {
window.scheduler.postTask(() => {
initializeGTM();
}, { priority: 'background' });
}, 10000);
} else {
requestIdleCallback(() => {
initializeGTM();
}, { timeout: 8000 });
}
}

function initializeGTM() {
// GTM initialization code
(!function(e,t,n,o,a){
e[o]=e[o]||[];
e[o].push({"gtm.start":(new Date).getTime(),event:"gtm.js"});
var i=t.getElementsByTagName(n)[0],c=t.createElement(n),d="dataLayer"!=o?"&l="+o:"";
c.defer=!0;
c.src="https://www.googletagmanager.com/gtm.js?id="+a+d;
i.parentNode.insertBefore(c,i);
}(window,document,"script","dataLayer",false ? "GTM-WXXDPPBN" : "GTM-5FCGK6"));

// Load GPT script with delay
setTimeout(() => {
lazyLoadJs(document,"script",["https://www.googletagservices.com/tag/js/gpt.js"]);
}, 2000);
}

// Optimized user interaction handler
function handleUserInteraction() {
if (gtmInitialized) return;

window.dispatchEvent(new CustomEvent('custom:userHasInteracted'));
window.userInteracted = true;

// Use scheduler API for better performance
if ('scheduler' in window && 'postTask' in window.scheduler) {
window.scheduler.postTask(() => {
scheduleGTMInit();
}, { priority: 'user-visible' });
} else {
// Fallback with minimal delay
setTimeout(() => {
scheduleGTMInit();
}, 10);
}
}

// Attach optimized event listeners
document.body.addEventListener('click', handleUserInteraction, { passive: true });
window.addEventListener('scroll', handleUserInteraction, { passive: true });

window.addEventListener('message', receiveMessage, false);

function receiveMessage(event) {
if(event.data =='close_dfp'){
if (typeof closeBanner1 === 'function') {
setTimeout(()=>{
closeBanner1();
},0)
} else {
console.info('closeBanner1 is not a function');
}
}
}

if('0' == 1){
window.pushNotiConf = {
apiDomain : 'https://apis.shiksha.com',
deviceType : 'desktop',
fromPWA : true
};

function loadBNDependency(){if(null!=document.getElementById("pushNotificationContainer")){var e=new Date,t=e.getDate()+""+e.getMonth()+e.getFullYear()+"v10",o=Math.floor(100*Math.random()),n=function(){var e=document.createElement("script");e.src="https://js.shiksha.ws/pwa/public/unavoidable_js/shikshaPushNotification.js?"+t+o,e.async=!0,document.head.appendChild(e)},c=document.createElement("script");c.src="https://js.shiksha.ws/pwa/public/push-notification-helpers.js?"+t+o,c.async=!0,document.head.appendChild(c),"object"==typeof c.onload?c.onload=function(e){console.log("notification helper sdk loaded"),n()}:n()}}
}

window.disableInstant = true;

if (typeof history !== 'undefined' && 'scrollRestoration' in history) {
history.scrollRestoration = 'manual';
}

var spanForPDF = document.createElement('span');
spanForPDF.id = 'pageCSSPathForPDF';
spanForPDF.style = 'display:none;';
var att = document.createAttribute("csspath");
spanForPDF.setAttributeNode(att);
document.body.appendChild(spanForPDF);

function loadMultipleCSS(cssArrayList){
if(cssArrayList && cssArrayList.length>0){
cssArrayList.forEach(function(url) {
var script = document.createElement('link');
script.href = url;
script.rel = 'stylesheet';
script.type = 'text/css';
document.body.appendChild(script);
});
}
}

function loadMultipleScriptOneByOne(scriptArayList){
return new Promise( function( resolve, reject ) {
var listItem = scriptArayList.shift();
if(listItem){
var s = document.createElement('script');
var r = false;
s.type = 'text/javascript';
s.src = listItem;
s.async = true;
s.onerror = function(err) {
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
reject(err, s);
}
};
s.onload = s.onreadystatechange = function() {
// console.log(this.readyState); // uncomment this line to see which ready states are called.
if (!r && (!this.readyState || this.readyState == 'complete')) {
r = true;
if(scriptArayList.length > 0 ) {
loadMultipleScriptOneByOne(scriptArayList).then( function() {
resolve();
});
} else {
resolve();
}
}
};
var t = document.getElementsByTagName('script')[0];
t.parentElement.insertBefore(s, t);
}else{
resolve();
}
});
}
function openChatBotRegistration(paramsObject,callBackFunction,onClose){
//versioning the files
var eVr=new Date();
var tVr= eVr.getDate()+""+eVr.getMonth()+eVr.getFullYear()+"081026";
var cssArray = ['https://js.shiksha.ws/public/css/convMMP.css?v='+tVr];
//var jsArray = ['https://js.shiksha.ws/public/js/jquery-3.5.1.min.js?v='+tVr,'https://js.shiksha.ws/public/js/vue.js?v='+tVr,'https://js.shiksha.ws/public/js/chatui.js?v='+tVr,'https://js.shiksha.ws/public/js/convMMP.js?v='+tVr];
var jsArray = ['https://js.shiksha.ws/public/js/botui-core.js?v='+tVr, 'https://js.shiksha.ws/public/js/convMMP.js?v='+tVr];
// GTM registration events live in PWA (window.pushShikshaChatbotRegistrationGtm from client bundle), not convMMP.js
var registrationCallback = (typeof callBackFunction === 'function')
? function(responseData) {
if (responseData && responseData.status === 'SUCCESS' && typeof window.pushShikshaChatbotRegistrationGtm === 'function') {
window.pushShikshaChatbotRegistrationGtm(responseData);
}
callBackFunction(responseData);
}
: callBackFunction;
// && paramsObject.entityId > 0
if(typeof(paramsObject) == 'object' && paramsObject !== null && (paramsObject.entityId > 0 || paramsObject.allowNullContId == true )){
if(typeof(loadConversationalMMP) === 'function'){
loadConversationalMMP(paramsObject,registrationCallback,onClose);
}else{
loadMultipleCSS(cssArray);
loadMultipleScriptOneByOne(jsArray).then(function (res) {
loadConversationalMMP(paramsObject,registrationCallback,onClose);
})
}
}

}
// window.blockDfpConfig = {'examPageMain': [21, 23], 'articlePage': []};
window.blockDfpConfig = {'examPageMain': [], 'articlePage': []};
window.shouldShowPEQ = true;
window.shouldShowDRC = true;
window.shouldEnableEventTracking = true;
window.shouldEnableExperimentalDFPEventTracking = false;
window.shouldShowTruecallerLayer = true;
window.shouldEnableMetaPixelFunnelSignalLogging = true;<iframe src="https://www.googletagmanager.com/ns.html?id=GTM-5FCGK6"
height="0" width="0" style="display:none;visibility:hidden"></iframe>let fbqInitialized = false;

function initializeFBQ() {
if (fbqInitialized) return;
fbqInitialized = true;

!function(f,b,e,v,n,t,s){if(f.fbq)return;n=f.fbq=function(){n.callMethod?
n.callMethod.apply(n,arguments):n.queue.push(arguments)};if(!f._fbq)f._fbq=n;
n.push=n;n.loaded=!0;n.version='2.0';n.queue=[];t=b.createElement(e);t.async=!0;
t.src=v;s=b.getElementsByTagName(e)[0];s.parentNode.appendChild(t,s)}(window,
document,'script','https://connect.facebook.net/en_US/fbevents.js');
fbq('init', '639671932819149');
fbq('track', "PageView");
}

// Check if user has already interacted
if (window.userInteracted) {
// User has already interacted, initialize Facebook Pixel immediately
initializeFBQ();
} else {
// Wait for user interaction
window.addEventListener('custom:userHasInteracted', initializeFBQ);
}<img height="1" width="1" style="display:none"
src="https://www.facebook.com/tr?id=639671932819149&ev=PageView&noscript=1"
/>[1033,14830,81059,78002,68395,10194,96179,52052,8059,58539,73342,62097,56227,48003,65024,34680,78955,46958,53146,71355,94995,5924,89734,54997,35675,42397,93650]{"namedChunks":["desktop_header","TagDetailPage","AskHomeDiscussionCard","SocialSharingTooltip","VoteComponent","Icons","RegisterPromptWidget","TdpMostActiveUsers","ReviewWidget","desktop_footer","ShikshaCommonFooter"]}function stopHashScroll(event) {
event.preventDefault();
window.scrollTo(0, 0); // Scroll back to the top
}
// Function to re-enable scrolling
function enableHashScroll() {
window.removeEventListener("scroll", stopHashScroll);
}
const hash = window.location.hash;
if(hash) {
// Prevent the default scrolling behavior when the page has an anchor in the URL
window.addEventListener("scroll", stopHashScroll);
// Enable scroll when the page loads
window.addEventListener("load", enableHashScroll);
}window.dataLayer = window.dataLayer || [];
window.gtag = function (){dataLayer.push(arguments);}
window.gtag('js', new Date());
window.gtag('config', "G-K11YYW39J9");}{l i m}$

Limits and Derivatives

Questions

Discussions

Active Users

Followers

The sum of the roots of the equation, $x + 1 - 2 l o g_{2} (3 + 2^{x}) + 2 l o g_{4} (10 - 2^{- x}) = 0,$ is :

42. Find the derivative of the following functions:

(i)sin x cos x (ii) $s e c x$ (iii)5 sec x + 4 cosx

(iv) $c o s e c x$ (v)3cot x + 5 cosec x

(vi) $5 s i n x - 6 c o s x + 7$ (vii) $2 t a n x - 7 s e c x$

29. Kindly consider the following

17. Kindly consider the following

18. $\underset{x \to 0}{l i m} \frac{a x + x c o s x}{b s i n x}$

Kindly consider the following

73. $\frac{x}{s i n^{n} x}$

Kindly consider the following

72. (x + sec x) (x- tan x

Kindly consider the following

71. $\frac{x}{1 + t a n x}$

Kindly consider the following

70. $\frac{x^{2} c o s (\frac{π}{4})}{s i n x}$

Kindly consider the following

69. $\frac{4 x + 5 s i n x}{3 x + 7 c o s x}$