Mathematics - Differentiation Question with Solution | TestHub

Given function can be written as

$y={\tan }^{-1}\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$

$\Rightarrow y={\tan }^{-1}\left(\frac{\tan x-1}{\tan x+1}\right)$

$\Rightarrow y={\tan }^{-1}\left(\tan \left(x-\frac{\pi }{4}\right)\right) \left(\mathrm{given} x\in \left(0,\frac{\pi }{2}\right)\right)$
$\Rightarrow y=x-\frac{\pi }{4}$
Now $\frac{dy}{d\left(\frac{x}{2}\right)}=2\frac{dy}{dx}=2$