Question

Find the value of $\frac{\frac{d}{dx}{e}^x(\frac{x-2}{x+2}{)}^{3/4}}{e^x(\frac{x-2}{x+2}{)}^{3/4}}$

• A. $\frac{x^2-1}{x^2-4}$
• B. $1$
• C. $\frac{x^2+1}{x^2-4}$
• D. $\frac{x^2-1}{x^2+4}$

Answer 1

The correct option is A $\frac{x^2-1}{x^2-4}$

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

Taking log on both sides

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

Differentiating w.r.t. $x$, we get,

$\frac{1}{y}\frac{dy}{dx}=1+\frac{3}{4}\left(\frac{x+2}{x-2}\right)\left(\frac{(x+2)-(x-2)}{(x+2{)}^2}\right)$

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

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