Differentiate the following with respect to x: ^ -1 ( x) - Vidyadip

Question

Differentiate the following with respect to x:
$\cos^{-1}(\sin\text{ x})$

✓

Answer

Let $\text{f(x)}=\cos^{-1}(\sin\text{x})$
We observe that this function is defined for all real numbers.
$\text{f(x)}=\cos^{-1}(\sin\text{x})$
$=\cos^{-1}\Big[\cos\Big(\frac{\pi}{2}-\text{x}\Big)\Big]=\frac{\pi}{2}-\text{x}$
Thus, $\text{f(x)}=\frac{\text{d}}{\text{dx}}\Big(\frac{\pi}{2}-\text{x}\Big)=-1$

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