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# Solution for Differentiate X Sin − 1 X ? - CBSE (Commerce) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate $x^{\sin^{- 1} x}$  ?

#### Solution

$\text{ Let y } = x^{\sin^{- 1} x} . . . \left( i \right)$
$\text{ Taking log on both sides },$
$\log y = \log x^{\sin^{- 1} x }$
$\Rightarrow \log y = \sin^{- 1} x \log x$
$\text{ Differentiating with respect to x },$
$\frac{1}{y}\frac{dy}{dx} = \sin^{- 1} x\frac{d}{dx}\left( \log x \right) + \left( \log x \right)\frac{d}{dx}\left( \sin^{- 1} x \right)$
$\Rightarrow \frac{1}{y}\frac{dy}{dx} = \sin^{- 1} x\left( \frac{1}{x} \right) + \left( \log x \right)\left( \frac{1}{\sqrt{1 - x^2}} \right)$
$\Rightarrow \frac{dy}{dx} = y\left[ \frac{\sin^{- 1} x}{x} + \frac{\log x}{\sqrt{1 - x^2}} \right]$
$\Rightarrow \frac{dy}{dx} = x^{\sin^{- 1} x} \left[ \frac{\sin^{- 1} x}{x} + \frac{\log x}{\sqrt{1 - x^2}} \right] \left[ \text{ using equation } \left( i \right) \right]$

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Solution Differentiate X Sin − 1 X ? Concept: Simple Problems on Applications of Derivatives.
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