Question

Differentiate $\frac{d{5}^{\left(e^x{\sin }^{2}x\right)}}{dx}$

Answer 1

let $y=5^{e^{x}si{n}^{2}x}$

apply log on both sides

$logy=\left(e^{x}si{n}^{2}x\right)log5$

now differentiate on both sides with respect to x

$\frac{d}{dx}\left(logy\right)=log5\frac{d}{dx}{e}^{x}si{n}^{2}x$

$\frac{y^{!}}{y}=log5(e^{x}si{n}^{2}x+e^{x}sin2x)$

therefore $\frac{dy}{dx}=(e^{x}si{n}^{2}x+e^{x}sin2x)5^{e^{x}si{n}^{2}x}log5$