Differentiate the following w. r. t. x. ^ -1 (2^x ) - Vidyadip

Question

Differentiate the following w. r. t. x.$\sin ^{-1}\left(2^x\right)$

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Answer

Let $y=\sin ^{-1}\left(2^x\right)$
Differentiate $w . r . t . x$.
$
\begin{aligned}
\frac{d y}{d x} & =\frac{d}{d x}\left(\sin ^{-1}\left(2^x\right)\right) \\
& =\frac{1}{\sqrt{1-\left(2^x\right)^2}} \cdot \frac{d}{d x}\left(2^x\right) \\
& =\frac{1}{\sqrt{1-2^{2 x}}}\left(2^x \log 2\right) \\
\therefore \quad \frac{d y}{d x} & =\frac{2^x \log 2}{\sqrt{1-4^x}}
\end{aligned}
$

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