Limits and Derivatives

Get insights from 93 questions on Limits and Derivatives, answered by students, alumni, and experts. You may also ask and answer any question you like about Limits and Derivatives

93
Questions
0
Discussions
0
Active Users
0
Followers

Have a question about Limits and Derivatives? Ask our experts

Ask Your Question

Keep it short & simple. Type complete word. Avoid abusive language.

New answer posted•a year ago

Limits and Derivatives Limits and Derivatives

48.

Kindly consider the following

$\frac{a x + b}{c x + d}$

5 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

48. Given, f (x) = $\frac{a x + b}{c x + d}$

So, f (x) = $\frac{(c x + d) \frac{d}{d x} (a x + b) - (a x + b) \frac{d}{d x} (c x + d)}{({(x + d)}^{2}}$

$= \frac{(c x + d) a - (a x + b) c}{{(c x + d)}^{2}}$

$= \frac{a c x + a d - a c x - c b}{{(c x + d)}^{2}}$

$= \frac{a d - b c}{{(c x + d)}^{2}}$

Upvote

New question posted•a year ago

Limits and Derivatives Limits and Derivatives

<math><mrow><munder><mrow><mi>l</mi><mi>i</mi><mi>m</mi></mrow><mrow><mi>x</mi><mo>→</mo><mn>3</mn></mrow></munder></mrow></math> x+ 3

1 Views|Posted a year ago

Asked by Alok Kumar Singh

New answer posted•a year ago

Limits and Derivatives Limits and Derivatives

47. (ax + b) (cx + d)²

6 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

47. Given, f (x) = (ax + b) (cx + d)²

So, f?(x) = (ax +b) $\frac{d}{d x} {(c x + d)}^{2} + {(c x + d)}^{2} \frac{d}{d x} (a x + b)$

$= (a x + b) \frac{d}{d x} [(c + d) (c x + d)] + {(c x + d)}^{2} a$

$= (a x + b) [(c + d) \frac{d}{d x} (x + d) + (c + d) \frac{d}{d x} (c + d)] + a {(c x + d)}^{2}$

$= (a x + b) [c (c x + d) + c (c x + d)] + a {(c x + d)}^{2}$

$= (a x + b) \cdot 2 \cdot c (c x + d) + a {(c x + d)}^{2}$

Upvote

New answer posted•a year ago

Limits and Derivatives Limits and Derivatives

44. Find the derivative of the following functions from first principle:

(i) x (ii) (x)1 (iii) sin(x + 1) (iv) cos $(x - \frac{π}{8})$

Find the derivative of the following functions (it is to be understood that a, b, c, d,

p, q, r and s are fixed non-zero constants and m and n are integers):

23 Views|Posted a year ago

Asked by Alok Kumar Singh

Answered by

Alok Kumar Singh | Contributor-Level 10

a year ago

44.

= \underset{h \to 0}{l i m} \frac{- x - h + x}{h}

= \underset{h \to 0}{l i m} \frac{- h}{h}

= \underset{h \to 0}{l i m} - 1

= - 1.

(ii) Given, f(x) = (-x)^-¹

by first principle,

f(x) $\underset{h \to 0}{l i m} \frac{[-] {(x + h)}^{- 1} - {(- x)}^{- 1}}{h}$

(iii) Given, f(x) = sin(x + 1)

By first principle,

f’(x) = $\underset{h \to 0}{l i m} \frac{f (x + h) - f (x)}{h}$

$= \underset{h \to 0}{l i m} \frac{s i n (x + h + 1) - s i n (x + 1)}{h}$

$= \underset{h \to 0}{l i m} \frac{2}{h} \cdot c o s (\frac{x + h + 1 + x + 1}{2}) \cdot s i n (\frac{x + h + 1 - (x + 1)}{2})$

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

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

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

= cos (x + 1)

(iv) Given, f(x) = cos $(x - \frac{π}{8})$

By first principle,

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

$= \underset{h \to 0}{l i m} \frac{c o s (x + h - \frac{π}{8}) - c o s (x - \frac{π}{8})}{h}$

Upvote

New answer posted•a year ago

Limits and Derivatives Limits and Derivatives

43. 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) $2 t a n x - 7 s e c x$

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

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

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

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

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

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

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

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

$= \frac{3}{s i n x ? s i n x} \times (? 1) + 5 l i m$

Upvote

New answer posted•a year ago

Limits and Derivatives Limits and Derivatives

43. 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) $2 t a n x - 7 s e c x$

5 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