Differentiate the function ( x)^ x + ^ -1 x with respect to x. - Vidyadip

Question

Differentiate the function $(\sin x)^{x} + \sin^{-1} \sqrt{x}$ with respect to x.

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Answer

$\text{y} = (\sin \text{x})^{\text{x}} + \sin^{-1} \sqrt{\text{x}}$
$\text{y = u + v} \Rightarrow \frac{\text{dy}}{\text{dx}} = \frac{\text{du}}{\text{dx}} + \frac{\text{dv}}{\text{dx}}$
$\text{u} = (\sin \text{x})^{\text{x}}$
$\Rightarrow \log \text{u = x } \log \sin \text{x} $
$\Rightarrow \frac{\text{du}}{\text{dx}} = (\sin \text{x})^{\text{x}} [\text{x} \cot \text{x} + \log \sin \text{x}]$
$\text{v} = \sin^{-1} \sqrt{\text{x}}$
$\Rightarrow \frac{\text{dv}}{\text{dx}} = \frac{1}{2\sqrt{\text{x - x}^{2}}}$
$\therefore \frac{\text{dy}}{\text{dx}} = (\sin \text{x})^{\text{x}} [\text{x} \cot \text{x} + \log \sin \text{x}] + \frac{1}{2\sqrt{\text{x - x}^{2}}}$

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