Find the  $\frac{dy}{dx}$ for $y=\log \left(e^x{\left(\frac{x-2}{x+2}\right)}^{\frac{3}{4}}\right)$

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

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

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

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

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

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

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