# Product Rule for Derivatives/Examples/2 a x times Exponential of a x^2

## Examples of Use of Product Rule for Derivatives

$\map {\dfrac \d {\d x} } {2 a x e^{a x^2} } = 2 a e^{a x^2} \paren {1 + 2 a x^2}$

## Proof

Let $u = 2 a x$.

Let $v = e^{a x^2}$.

Thus we have:

$y = u v$

and so:

 $\ds \dfrac {\d y} {\d x}$ $=$ $\ds v \dfrac {\d u} {\d x} + u \dfrac {\d v} {\d x}$ Product Rule for Derivatives $\ds$ $=$ $\ds e^{a x^2} \cdot 2 a + 2 a x \cdot \paren {2 a x e^{a x^2} }$ Power Rule for Derivatives, Derivative of $e^{a x^2}$ $\ds$ $=$ $\ds 2 a e^{a x^2} \paren {1 + 2 a x^2}$ simplification

$\blacksquare$