Differentiate the following function w.r.t. x:X^ x + ( x)^ x - Vidyadip

Question

Differentiate the following function w.r.t. x:$X^{\sin x} + (\sin x)^{\cos x} $

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Answer

$\text{Let x}^{\sin x} = \text{u, and} ( \sin \text{x)}^{\cos \text{x}} = \text{v} \therefore \text{y = u + v} \Rightarrow \frac{\text {dy}}{\text{dx}} = \frac{\text{du}}{\text{dx}} + \frac{\text{dv}}{\text{dx}}$
$\text{Getting} \frac{\text{du}}{\text{dx}} = \text{x}^{\sin{\text{x}}} \bigg[\frac{\sin\text{x}}{\text{x}} + \log \text{x} . \cos{\text{x}} \bigg]$
$\text{and} \frac{\text{dv}}{\text{dx}} = ( \sin \text{x})^{\cos\text{x}}[\cos\text{x} .\cot\text{x} - \sin\text{x} \log \sin \text{x}]$
$\therefore \frac{\text{dy}}{\text{dx}} = \text{x}^{\sin \text{x}} \bigg[ \frac{\sin \text{x}}{\text{x}} + \log\text{x}.\cos\text{x}\bigg] + (\sin\text{x}) ^{\cos{\text{x}}} [\cos\text{x}. \cot\text{x} -\sin\text{x} \log\sin\text{x}]$

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