Differentiate the ^ -1 ( x 1+ x ) w.r.t. x. - Vidyadip

Question

Differentiate the $\tan ^{-1}\left(\frac{\sin x}{1+\cos x}\right) \text{w.r.t. x}.$

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Answer

Let $f(x) = \tan^{–1}\left(\frac{\sin x}{1+\cos x}\right)$.
Observe that this function is defined for all real numbers, where $\cos x \neq – 1;$
i.e., at all odd multiples of $\pi$.
We may rewrite this function as
$f(x) = \tan^{-1}\left(\frac{\sin x}{1+\cos x}\right)$
$= \tan^{-1}\left[\frac{2 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{2}\right)}{2 \cos ^{2} \frac{x}{2}}\right]$
$f(x) = \tan^{-1}\left[\tan \left(\frac{x}{2}\right)\right]=\frac{x}{2}$
$\Rightarrow f^\prime(x)= \frac12$

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