Question

Find $\frac{dy}{dx}$ where $y=\log \left(\sec x\right)0\le x\le \frac{\pi }{2}$

Answer 1

$0\le x\le \frac{\pi }{2}$

$\sec x$ is positive

Hence, the function $y$ is defined on that interval

$y=\log \left(\sec x\right)$

Differentiate with respect to $x$ on both sides

$⟹\frac{dy}{dx}=\frac{1}{\sec x}\frac{d}{dx}\left(\sec x\right)⟹\frac{dy}{dx}=\frac{1}{\sec x}\tan x\sec x$

$⟹\frac{dy}{dx}=\tan x$