Question

Find $\frac{dy}{dx}$, if $y=\log \left(e^x{\sin }^{5}x\right)$

Answer 1

Given, $y=\log \left(e^x{\sin }^{5}x\right)$

On differentiating w.r.t. $x$, we get

$\frac{dy}{dx}=\frac{1}{e^x{\sin }^{5}x}\cdot \frac{d}{dx}\left(e^x{\sin }^{5}x\right)$

$=\frac{1}{e^x{\sin }^{5}x}[e^x\cdot 5{\sin }^{4}x\cdot \cos x+{\sin }^{5}x\cdot e^x]$

$=\frac{{\sin }^{4}x{e}^x}{e^x{\sin }^{5}x}[5\cos x+\sin x]$

$=\frac{5\cos x+\sin x}{\sin x}$

$=5\cot x+1$