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Question: \(\frac{d}{dx}\left\lbrack \log\left\{ e^{x}\left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right...

ddx[log⁡{ex(x−2x+2)3/4}]\frac{d}{dx}\left\lbrack \log\left\{ e^{x}\left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right\rbrack equals to

A

11

B

x2+1x2−4\frac{x^{2} + 1}{x^{2} - 4}

C

x2−1x2−4\frac{x^{2} - 1}{x^{2} - 4}

D

exx2−1x2−4e^{x}\frac{x^{2} - 1}{x^{2} - 4}

Answer

x2−1x2−4\frac{x^{2} - 1}{x^{2} - 4}

Explanation

Solution

Lety=[log⁡{ex(x−2x+2)3/4}]=log⁡ex+log⁡(x−2x+2)3/4y = \left\lbrack \log\left\{ e^{x}\left( \frac{x - 2}{x + 2} \right)^{3/4} \right\} \right\rbrack = \log e^{x} + {\log\left( \frac{x - 2}{x + 2} \right)}^{3/4}

⇒ y=x+34[log⁡(x−2)−log⁡(x+2)]y = x + \frac{3}{4}\lbrack\log(x - 2) - \log(x + 2)\rbrack

⇒ dydx=1+34[1x−2−1x+2]=1+3(x2−4)\frac{dy}{dx} = 1 + \frac{3}{4}\left\lbrack \frac{1}{x - 2} - \frac{1}{x + 2} \right\rbrack = 1 + \frac{3}{(x^{2} - 4)}⇒ dydx=x2−1x2−4\frac{dy}{dx} = \frac{x^{2} - 1}{x^{2} - 4}.