Question

Differentiate $y=\frac{\sin x}{e^x}$ w.r.t $x$

Answer 1

$y=\frac{\sin x}{e^x}$

$\Rightarrow y{e}^x=\sin x$

$\Rightarrow y\frac{d}{dx}\left(e^x\right)+e^x\frac{dy}{dx}=\frac{d}{dx}\left(\sin x\right)$

$\Rightarrow y{e}^x+e^x\frac{dy}{dx}=\cos x$

$\Rightarrow e^x\frac{dy}{dx}=\cos x-y{e}^x$

$\Rightarrow \frac{dy}{dx}=\frac{\cos x-y{e}^x}{e^x}$

$\therefore \frac{dy}{dx}=e^{-x}\cos x-y$