Question

If y = log (cos e^x), then find $\frac{dy}{dx}.$

Answer 1

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

Differentiating both sides with respect to x, we get

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

$\Rightarrow \frac{dy}{dx}=\frac{1}{\cos e^x}\times \frac{d}{dx}\cos e^x$

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

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

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