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

Question

Differentiate the following function with respect to $x:x :(\cos x)^x ;\left(\right.$ where $\left.x \in\left(0, \frac{\pi}{2}\right)\right)$

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Answer

Let $y =(\cos x)^x$.
Then, $y =e^{x \log _e \cos x}$ On differentiating both sides with respect to $x$,
we get $\frac{d y}{d x}=e^{x \log _e \cos x} \frac{d}{d x}\left(x \log _e \cos x\right)$
$ \Rightarrow \frac{d y}{d x}=(\cos x)^x\left\{\log _e \cos x \frac{d}{d x}(x)+x \frac{d}{d x}\left(\log _e \cos x\right)\right\}$
$\Rightarrow \frac{d y}{d x}=(\cos x)^x\left\{\log _e \cos x+x \cdot \frac{1}{\cos x}(-\sin x)\right\} $
$\Rightarrow \frac{d y}{d x}=(\cos x)^x\left(\log _e \cos x-x \tan x\right)$

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