Differentiate the following w.r.t.x: ( +e^ x ),x>0 - Vidyadip

Question

Differentiate the following w.r.t.x: $\cos({\log\text{x}}+\text{e}^{\text{x}}),\text{x}>0$

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Answer

$\text{Let y}=\cos({\log\text{x}}+\text{e}^{\text{x}})$
$\therefore\ \frac{\text{dy}}{\text{dx}} =-\sin(\log\text{x}+\text{e}^\text{x})\frac{\text{d}}{\text{dx}}(\log\text{x}+\text{e}^\text{x})$
$ =-\sin(\log\text{x}+\text{e}^\text{x}).\Big(\frac{1}{\text{x}}+\text{e}^\text{x}\Big)$
$ =-\Big(\frac{1}{\text{x}}+\text{e}^\text{x}\Big)\sin(\log\text{x}+\text{e}^\text{x})$

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