Differentiate x^ sin x + ( x)^ x with respect to x. - Vidyadip

Question

Differentiate $x^{sin x} + (\sin x)^{\cos x}$ with respect to $x.$

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Answer

$\text{let y = u + v, u = x}^{\sin\text{x}}, \text{v }={(\sin \text{x})}^{\cos \text{x}} $
$\log \text{u} = \sin \text{x}\log \text{x}\Rightarrow\frac{\text{du}}{\text{dx}} = \text{x}^{\sin \text{x}}.\left\{\cos\text{x}\log\text{x} + \frac{\sin\text{x}}{\text{x}}\right\}$
$\log \text{v} = \cos \text{x}.\log(\sin \text{x})\Rightarrow\frac{\text{dv}}{\text{dx}} = (\sin\text{x})^{\cos \text{x}}. \left\{\cos \text{x}. \cot\text{x} - \sin\text{x}. \log(\sin \text{x})\right\}$
$\frac{\text{dy}}{\text{dx}}= \frac{\text{du}}{\text{dx}} + \frac{\text{dv}}{\text{dx}} = \text{x}^{\sin \text{x}}.\left\{ \cos\text{x}.\log\text{x} + \frac{\sin\text{x}}{\text{x}} + (\sin\text{x})^{\cos\text{x}}\right\}\left\{\cos \text{x}.\cot \text{x} - \sin\text{x}. \log (\sin \text{x})\right\}$

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