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

Question

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

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Answer

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

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