d dx [ e^x ( x - 2 x + 2 ) ^ 3/4 ] is equals to - Vidyadip

MCQ

${d \over {dx}}\left[ {\log \left\{ {{e^x}{{\left( {{{x - 2} \over {x + 2}}} \right)}^{3/4}}} \right\}} \right]$ is equals to

A
$1$
B
${{{x^2} + 1} \over {{x^2} - 4}}$
✓
${{{x^2} - 1} \over {{x^2} - 4}}$
D
${e^x}{{{x^2} - 1} \over {{x^2} - 4}}$

✓

Answer

Correct option: C.

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

c
(c) Let $y = \left[ {\log \left\{ {{e^x}{{\left( {\frac{{x - 2}}{{x + 2}}} \right)}^{3/4}}} \right\}} \right] $

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

==> $y = x + \frac{3}{4}\,[\log (x - 2) - \log (x + 2)]$

==> $\frac{{dy}}{{dx}} = 1 + \frac{3}{4}\,\left[ {\frac{1}{{x - 2}} - \frac{1}{{x + 2}}} \right] $

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

==> $\frac{{dy}}{{dx}} = \frac{{{x^2} - 1}}{{{x^2} - 4}}$.

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