ddxlog⁡exx−2x+23/4 is

# $\frac{d}{dx}\left[\mathrm{log}{\left\{{e}^{x}\left(\frac{x-2}{x+2}\right)\right\}}^{3/4}\right]$ is

1. A

1

2. B

$\frac{{x}^{2}+1}{{x}^{2}-4}$

3. C

$\frac{{x}^{2}-1}{{x}^{2}-4}$

4. D

${e}^{x}\frac{{x}^{2}-1}{{x}^{2}-4}$

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### Solution:

let $y=\left[\mathrm{log}\left\{{e}^{x}{\left(\frac{x-2}{x+2}\right)}^{3/4}\right\}\right]$
$\begin{array}{l}=\mathrm{log}{e}^{x}+\mathrm{log}{\left(\frac{x-2}{x+2}\right)}^{3/4}\\ ⇒y=x+\frac{3}{4}\left[\mathrm{log}\left(x-2\right)-\mathrm{log}\left(x+2\right)\right]\end{array}$
On differentiating w.r.t. x, we get
$\begin{array}{l}\frac{dy}{dx}=\frac{d}{dx}\left[x+\frac{3}{4}\left\{\mathrm{log}\left(x-2\right)-\mathrm{log}\left(x+2\right)\right\}\right]\\ =1+\frac{3}{4}\left[\frac{1}{x-2}-\frac{1}{x+2}\right]=1+\frac{3}{{x}^{2}-4}⇒\frac{dy}{dx}=\frac{{x}^{2}-1}{{x}^{2}-4}\end{array}$

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