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

Question

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

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Answer

Let $\text{y}=\sin^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big)$
Put $\text{x}=\cot\theta$
$\text{y}=\sin^{-1}\Big(\frac{1}{\sqrt{1+\cot^2\theta}}\Big)$
$=\sin^{-1}\Big(\frac{1}{\sqrt{\text{cosec}^2\theta}}\Big)$
$=\sin^{-1}(\sin\theta)$
$=\theta$
$\text{y}=\cot^{-1}\text{x}\ [\text{Since}, \cot\theta=\text{x}]$
Differentiating it with respect to x,
$\frac{\text{dy}}{\text{dx}}=-\frac{1}{(1+\text{x}^2)}$

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