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

Question

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

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Answer

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

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