Question

If $y=(\sin x{)}^{\log x}$ then find $\frac{dy}{dx}$.

Answer 1

$y=(\sin x{)}^{\log x}$

$\log \left(y\right)=\log x.\log \left(\sin x\right)$

$\frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\log \left(\sin x\right)+\frac{\log x}{\sin x}\times \cos x$

$\frac{1}{y}\frac{dy}{dx}=\frac{\log \left(\sin x\right)}{x}+\frac{\log x}{\tan x}$

$\frac{dy}{dx}=(\sin x{)}^{\log x}[\frac{\log \left(\sin x\right)}{x}+\frac{\log x}{\sin x}]$