Differentiate the following functions with respect to x: (2 ^ -1 … - Vidyadip

Question

Differentiate the following functions with respect to x:
$\sin(2\sin^{-1}\text{x})$

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Answer

Let, $\text{y}=\sin(2\sin^{-1}\text{x})$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big(\sin(2\sin^{-1}\text{x})\Big)$
$=\cos\big(2\sin^{-1}\text{x}\big)\frac{\text{d}}{\text{dx}}\big(2\sin^{-1}\text{x}\big)$
[Using chain rule]
$=\cos\big(2\sin^{-1}\text{x}\big)\times2\frac{1}{\sqrt{1-\text{x}^2}}$
$=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\sin\big(2\sin^{-1}\text{x}\big)\Big)=\frac{2\cos\big(2\sin^{-1}\text{x}\big)}{\sqrt{1-\text{x}^2}}$

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