Differentiate the log( e^x) w.r.t. x. - Vidyadip

Question

Differentiate the log$(\cos e^x)$ w.r.t. x.

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Answer

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

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