Application of Derivatives

(i) Let y= ?x$$ : Let x = 25 and x = 0. 3.

Then, ?y = ?x+?x$?x$

$=?25.3?25$

$=?25.3?5$

$?25.3=5+?y\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}\text{?}?y~dy.$

= 5 + dy

= $5+\left(\frac{dy}{dx}\right)Ax.$

$=5+\frac{1}{\mathrm{2?x}}?x.$

$=5+\frac{1}{\mathrm{2?25}}?0.3$

$=5+\frac{0.3}{2*5}$

= 5 + 0.03

$?25.3=5.030$

(ii) ?49.5$$

A.(ii)

Let y = ?x$.$ Let x = 49 and x = 0.5.

Then, $?y=?x+?x?x.$

$=?49.5?49$

$?49.5=7+?y=7+{\left(\frac{dy}{dx}\right)}_{}^{?x}$

$=7+\frac{1}{\mathrm{2?x}}?x$

$=7+\frac{1}{\mathrm{2?49}}*(0.5)$

$=7+\frac{0.5}{14}$

= 7 + 0.0357.

$?49.5=7?035$

(iii) ?0.6$$

A.(iii)

Let y = ?x$.$ Let x = 0.64 and ?x = 0.04.

Then, $?y=?x+Ax?x$

$=?0.64-0.04?0.64.$

$=?0.6?0.8$

$?0.6=0.8+?y=0.8+\left(\frac{dy}{dx}\right)?x$

$=0.8+\frac{1}{\mathrm{2?x}}?x$

$=0.8+\frac{1}{\mathrm{2?0.64}}*(?\text{?}0.04)$

$=0.8\frac{?0.04}{2*0.8}$

= 0.8 - 0.025.

= 0.775.

(iv) ${(0.009)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

A.(iv)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$ Let x = 0.008 and ?x = 0.00 1.

Then, ?y = ${[x+?x]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}?{[x]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$={(0.009)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}?{(0.008)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

${(0.009)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=0.2+?y=0.2+\frac{dy}{dx}??x$

$=0.2+\frac{1}{3{\left(x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}^2}?x$

$=0.2+\frac{1}{3{\left[{(0.008)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right]}^2}*(0.001)$

$=0.2+\frac{0.001}{3*{(0.0)}^2}=0.2+\frac{0.001}{0.12}$

= 0.2 + 0.0083.

= 0.208.

(v) ${(0.999)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}$

A.(v)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}.$ Let x = 1 and ?x = -0.001

Then, $?y={(x+?x)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}$

$={(0.999)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}?{(1)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}$

$?1{(0.999)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}=1+Ay$

$=1+\frac{dy}{dx}?x.$

$=1+\frac{1}{10x^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$10$}\right.}}*?x$

$=1?\frac{0.001}{10(.1^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$10$}\right.})}=1?\frac{0.001}{10}=1?0.0001$

= 0.999.

(vi) ${(15)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

A.(vi)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$ Then, x = 16 and ?x = 1.

Then, $?y={(x+Ax)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$

$={(15)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?{(16)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

$?{(15)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=2+Ay=2+\frac{dy}{dx}Ax$

$=2+\frac{1}{4x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}?(?1)$

$=4?\frac{1}{4{(16)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

$=2?\frac{1}{4*8}$

$=4?\frac{1}{32}$

$=\frac{64?1}{32}=\frac{63}{32}=1.968$

(vii) ${(26)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

A.(vii)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}.$ Let x = 27 and ?x = 1.

Then, $?y={(x+?x)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$={(26)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}?{(27)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$

$?{(26)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}=3+4y=3+\frac{dy}{dx}Ax=3+\frac{1}{3x^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}Ax,$

$=3?\frac{1}{3{(2F)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

$=3?\frac{1}{27}$

$=\frac{81?1}{27}=\frac{80}{27}=2.962.$

(viii) ${(255)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

A.(viii)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$ Let x = 256 and ?x = 1.

Then, $?y={(x+?x)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$

$={(255)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?{(256)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

$?{(255)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}={(256)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}+4y=4+\frac{dy}{dx}Ax.$

$=4+\frac{1}{4x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}?4x$

$=4\frac{?1}{4{(256)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

$=4?\frac{1}{256}$

$=\frac{1024?1}{256}$

$=\frac{1023}{256}=3.996$

(ix) ${(82)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

A.(ix)

Let $y=x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}.$ let x = 81 and ?x = 1.

Then, $?y={(x+4x)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

$={(82)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}?{(81)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}$

$7{(82)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}=3+4y=3+\frac{dy}{dx}?x$

$=3+\frac{1}{4x^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}??x$

$=3+\frac{1}{4{(81)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}}$

$=3+\frac{1}{4*3^3}$

$=3+\frac{1}{108}=\frac{324+1}{108}=\frac{325}{108}=3.001.$

(x) ${(401)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

A.(x)

Let y = $x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ = ?x$.$ Let x = 400 and ?x = 1.

Then, $?y={(x+?x)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}?x^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}.$

$?\text{?}?y={(401)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}?{(400)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$

$?\text{?}{(401)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=20\text{?}+?y=20+\frac{dy}{dx}?x$

$=20+\frac{1}{\mathrm{2?x}}?x$

$=20+\frac{1}{\mathrm{2?400}}$

$=20+\frac{1}{2*20}$

$=20+\frac{1}{40}$

$=\frac{800+1}{40}=\frac{801}{40}=20.025.$