Question

If $y={\left(\sin x\right)}^x$, find $\frac{dy}{dx}$

Answer 1

$y={\left(\sin x\right)}^x$

$\therefore \ln y=x\ln \left(\sin x\right)$

Taking derivative of both sides

$\frac{1}{y}\frac{dy}{dx}=\ln \left(\sin x\right).1+x.\frac{1}{\sin x}.\cot x$

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

$\therefore \frac{dy}{dx}={\left(\sin x\right)}^x\{\ln \left(\sin x\right)+x\cot x\}$