Differentiate the following w.r.t.x: ( ^ x ) - Vidyadip

Question

Differentiate the following w.r.t.x: $\log(\cos\text{e}^{\text{x}})$

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Answer

$\text{Let}\ \text{y}=\log(\cos\text{e}^{\text{x}})$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{1}{\cos\text{e}^{\text{x}}}\frac{\text{d}}{\text{dx}}(\cos\text{e}^\text{x})\ \ \bigg[\because\frac{\text{d}}{\text{dx}}\log\text{f(x)}=\frac{1}{\text{f(x)}}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
$=\frac{1}{\cos\text{e}^{\text{x}}}(-\sin\text{e}^\text{x})\frac{\text{d}}{\text{dx}}\text{e}^\text{x}= \bigg[\because\frac{\text{d}}{\text{dx}}\cos\text{f(x)}=-\sin\text{f(x)}\frac{\text{d}}{\text{dx}}\text{f(x)}\bigg]$
$=-(\tan\text{e}^\text{x})\text{e}^\text{x}=-\text{e}^\text{x}(\tan\text{e}^\text{x})$

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