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

Question

Differentiate the $\cos^{–1} (\sin x) \text{w.r.t. x}.$

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Answer

Let $f (x) = \cos^{–1} (\sin x)$.
Observe that this function is defined for all real numbers.
We may rewrite this function as
$f(x) = \cos^{–1} (\sin x)$
$= \cos^{-1}\left[\cos \left(\frac{\pi}{2}-x\right)\right]$
$= \frac{\pi}{2}-x$
Thus $f ′(x) = –1.$

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