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AP EAMCET Engineering 26 Apr 2017 Shift 1 Paper

Section: Mathematics

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Question : 66 of 160

Marks: +1, -0

For

x^2-4\neq 0,

\frac{d}{dx}\left[\log\left\{e^x\left(\frac{x-2}{x+2}\right)^{3/4}\right\}\right]\text{ at }x=3

is

1
$\frac{8}{5}$
2
$\frac{8e^3}{5}$

Solution:

Consider the given function.

f(x)=\log\left\{e^x\left(\frac{x-2}{x+2}\right)^{3/4}\right\}

It is simplified as,

f(x)=\log e^{x}+\log\left(\frac{x-2}{x+2}\right)^{\frac{3}{4}}

=x\log e+\frac{3}{4}\log\left(\frac{x-2}{x+2}\right)

=x+\frac{3}{4}\log\left(\frac{x-2}{x+2}\right)

Differentiate with respect to

x

.

\frac{d}{dx}(f(x))

=1+\frac{3}{4}\left[\frac{x+2}{x-2}\left\{\frac{(x+2)\cdot 1-(x-2)\cdot 1}{(x+2)^2}\right\}\right]

=1+\frac{3}{4}\left(\frac{4}{x^{2}-4}\right)

=1+\frac{3}{x^{2}-4}

\left.\frac{d}{dx}f(x)\right|_{x=3}=1+\frac{3}{5}

Solve further,

\left.\frac{d}{dx}f(x)\right|_{x=3}=\frac{8}{5}

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