# What is the derivative of e^sin(x^2)?

Nov 18, 2016

$2 x {e}^{\sin \left({x}^{2}\right)} \cos \left({x}^{2}\right)$

#### Explanation:

$f \left(x\right) = {e}^{\sin \left({x}^{2}\right)}$

$f \left(x\right) = {e}^{g \left(x\right)}$ Where $g \left(x\right) = \sin \left({x}^{2}\right)$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} {e}^{g \left(x\right)} = {e}^{g \left(x\right)} \cdot g ' \left(x\right)$ By the Chain Rule

$f ' \left(x\right) = {e}^{\sin \left({x}^{2}\right)} \cdot \cos \left({x}^{2}\right) \cdot \frac{d}{\mathrm{dx}} {x}^{2}$ By the Chain Rule again

$f ' \left(x\right) = {e}^{\sin \left({x}^{2}\right)} \cdot \cos \left({x}^{2}\right) \cdot 2 x$

$f ' \left(x\right) = 2 x {e}^{\sin \left({x}^{2}\right)} \cos \left({x}^{2}\right)$