Question

Differentiate the given function w.r.t. $x$.
$y=\log \left(\cos e^x\right)$

Answer 1

Let $y=\log \left(\cos e^x\right)$

Thus using chain rule,

$\frac{dy}{dx}=\frac{d}{dx}\left[\log \right(\cos e^x\left)\right]$

$=\frac{1}{\cos e^x}.\frac{d}{dx}\left(\cos e^x\right)$

$=\frac{1}{\cos e^x}.(-\sin e^x).\frac{d}{dx}\left(e^x\right)$

$=\frac{-\sin e^x}{\cos e^x}.e^x$

$=-e^x\tan e^x,e^x\ne (2n+1)\frac{\pi }{2},n\ne N$