Question

Differentiate the following w.r.t. \(x\):
\(log(cos\,e^x)\)

Answer 1

The correct answer is \(=-e^xtan\,e^x,e^x≠(2n+1)\frac{π}{2},n∈N\)
Let \(y=log(cos\,e^x)\)
By using the chain rule,we obtain
\(\frac{dy}{dx}=\frac{d}{dx}[log(cos\,e^x)]\)
\(=\frac{1}{cos\,e^x}.\frac{d}{dx}(cos\,e^x)\)
\(=\frac{1}{cos\,e^x}.(-sin\,e^x).\frac{d}{dx}(e^x)\)
\(=\frac{-sin\,e^x}{cos\,e^x}.e^x\)
\(=-e^xtan\,e^x,e^x≠(2n+1)\frac{π}{2},n∈N\)