Differentiate the following functions with respect to x: x^ x - Vidyadip

Question

Differentiate the following functions with respect to x:
$\text{x}^{\sin{\text{x}}}$

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Answer

Let $\text{y}=\text{x}^{\sin{\text{x}}}\ .....(\text{i})$
Taking log on both the sides,
$\log\text{y}=\log\text{x}^{\sin{\text{x}}}$
$\log\text{y}=\sin\text{x}\log\text{x}\ \big[\text{Since,}\log\text{a}^\text{b}=\text{b}\log\text{a}\big]$
Differentiating with respect to x,
$\frac{1}{\text{y}}\frac{\text{dy}}{\text{dx}}=\sin\text{x}\frac{\text{d}}{\text{dx}}\log\text{x}+\log\text{x}\frac{\text{d}}{\text{dx}}\sin\text{x}$
[Using product rule]
$\frac{1}{\text{y}}\frac{\text{dt}}{\text{dx}}=\sin\text{x}\big(\frac{1}{\text{x}}\big)+\log\text{x}(\cos\text{x})$
$\frac{\text{dy}}{\text{dx}}=\text{y}\Big[\frac{\sin\text{x}}{\text{x}}+(\log\text{x})(\cos\text{x})\Big]$
Put the value of y,
$\frac{\text{dy}}{\text{dx}}=\text{x}^{\sin\text{x}}\Big[\frac{\sin\text{x}}{\text{x}}+(\log\text{x})(\cos\text{x})\Big]$

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