Differentiate cosec^ -1 ( 5^x ) w.r.t. x. - Vidyadip

Question

Differentiate $\operatorname{cosec}^{-1}\left(\sec 5^x\right)$ w.r.t. $x$.

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Answer

Let $y=\operatorname{cosec}^{-1}\left(\sec 5^x\right)$
$\therefore y=\operatorname{cosec}^{-1}\left[\operatorname{cosec}\left(\frac{\pi}{2}-5^x\right)\right]$
$\therefore y=\frac{\pi}{2}-5^x$
Differentiating w.r.t. $x$,
$\frac{d y}{d x}=0-5^x \log 5$
$=-5^x \log 5$

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