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# Solution for Differentiate ( Sin X ) Cos X ? - CBSE (Science) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate $\left( \sin x \right)^{\cos x}$ ?

#### Solution

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

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Solution Differentiate ( Sin X ) Cos X ? Concept: Simple Problems on Applications of Derivatives.
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