# How do you differentiate f(x)=sin(e^(3-x))  using the chain rule?

Apr 21, 2018

$f ' \left(x\right) = - {e}^{3 - x} \cos \left({e}^{3 - x}\right)$

#### Explanation:

The Chain Rule, when applied to the sine, tells us that

$\frac{d}{\mathrm{dx}} \sin \left(u\right) = \cos u \cdot \frac{\mathrm{du}}{\mathrm{dx}} ,$ where $u$ is some function in terms of $x .$

Here, $u = {e}^{3 - x}$, so

$f ' \left(x\right) = \cos \left({e}^{3 - x}\right) \cdot \frac{d}{\mathrm{dx}} {e}^{3 - x}$

$\frac{d}{\mathrm{dx}} {e}^{3 - x} = {e}^{3 - x} \cdot \frac{d}{\mathrm{dx}} \left(3 - x\right) = - {e}^{3 - x}$ so we get

$f ' \left(x\right) = - {e}^{3 - x} \cos \left({e}^{3 - x}\right)$