Question
Differentiate the following w. r. t. x.$y=\log \left[\cos \left(x^5\right)\right]$
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Answer
$y=\log \left[\cos \left(x^5\right)\right]$ Differentiate $w . r . t . x$ $ \begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}\left(\log \left[\cos \left(x^5\right)\right]\right) \\ & =\frac{1}{\cos \left(x^5\right)} \cdot \frac{d}{d x}\left[\cos \left(x^5\right)\right] \\ & =\frac{1}{\cos \left(x^5\right)}\left(-\sin \left(x^5\right)\right) \frac{d}{d x}\left(x^5\right) \\ \therefore \quad \frac{d y}{d x} & =-\tan \left(x^5\right)\left(5 x^4\right)=-5 x^4 \tan \left(x^5\right) \end{aligned} $
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