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

MCQ

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

✓
${{{x^2} - 7} \over {{x^2} - 4}}$
B
$1$
C
${{{x^2} + 1} \over {{x^2} - 4}}$
D
${e^x}{{{x^2} - 1} \over {{x^2} - 4}}$

✓

Answer

Correct option: A.

${{{x^2} - 7} \over {{x^2} - 4}}$

a
(a) $y = \log {e^x} + \frac{3}{4}\log \frac{{x + 2}}{{x - 2}} = x + \frac{3}{4}\log \frac{{x + 2}}{{x - 2}}$

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

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

$\frac{{dy}}{{dx}} = \frac{{{x^2} - 7}}{{{x^2} - 4}}$.

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