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

MCQ

$\frac{\text{d}}{\text{dx}}\bigg[\log\bigg\{\text{e}^\text{x}\Big(\frac{\text{x}-2}{\text{x}+2}\Big)^\frac{3}{4}\bigg\}\bigg]$ equals:

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

✓

Answer

Correct option: A.

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

Let, $\text{y}=\frac{\text{d}}{\text{dx}}\bigg[\log\bigg\{\text{e}^\text{x}\Big(\frac{\text{x}-2}{\text{x}+2}\Big)^\frac{3}{4}\bigg\}\bigg]$
$\Rightarrow\text{y}=\frac{\text{d}}{\text{dx}}\Big[\text{x}\log\text{e}+\frac{3}{4}\log\Big(\frac{\text{x}-2}{\text{x}+2}\Big)\Big]$
$\Rightarrow\text{y}=\frac{\text{d}}{\text{dx}}\Big[\text{x}+\frac{3}{4}\log\Big(\frac{\text{x}-2}{\text{x}+2}\Big)\Big]$
$\Rightarrow\frac{\text{d}}{\text{dx}}=1+\frac{3}{4\Big(\frac{\text{x}-2}{\text{x}+2}\Big)}\times\frac{(\text{x}+2)\times1-(\text{x}-2)\times1}{(\text{x}+2)^2}$
$\Rightarrow\frac{\text{d}}{\text{dx}}=1+\frac{3(\text{x}+2)}{4(\text{x}-2)}\times\frac{\text{x}+2-\text{x}+2}{(\text{x}+2)^2}$
$\Rightarrow\frac{\text{d}}{\text{dx}}=1+\frac{3(\text{x}+2)}{4(\text{x}-2)}\times\frac{4}{(\text{x}+2)}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=1+\frac{3}{(\text{x}^2-4)}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2-4+3}{\text{x}^2-4}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{\text{x}^2-1}{\text{x}^2-4}$

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