How do you differentiate f(x)=cos(e^(x) )  using the chain rule?

May 30, 2016

$\setminus \frac{d}{\mathrm{dx}} \cos \left({e}^{x}\right) = - \sin \left({e}^{x}\right) {e}^{x}$.

Explanation:

Chain rule says that

$\setminus \frac{d}{\mathrm{dx}} f \left[g \left(x\right)\right] = f ' \left[g \left(x\right)\right] g ' \left(x\right)$.

Our $f$ is the cosine and our $g$ is the exponential.
The derivative of $\cos$ is $- \sin$, while the derivative of the exponential is the exponential, so we have

$\setminus \frac{d}{\mathrm{dx}} \cos \left({e}^{x}\right) = - \sin \left({e}^{x}\right) {e}^{x}$.