# How do you differentiate f(x)= (4x^2+5)*e^(x^2) using the product rule?

Oct 18, 2016

$f ' \left(x\right) = 2 x {e}^{{x}^{2}} \left(4 {x}^{2} + 9\right)$

#### Explanation:

Product rule: $f ' \left(x\right) = u ' v + v ' u$

$f \left(x\right) = \left(4 {x}^{2} + 5\right) \cdot {e}^{{x}^{2}}$

Let $u = 4 {x}^{2} + 5$ and $v = {e}^{{x}^{2}}$

$u ' = 8 x$
$v ' = 2 x {e}^{{x}^{2}}$

$\therefore f ' \left(x\right) = 8 x \cdot {e}^{{x}^{2}} + 2 x {e}^{{x}^{2}} \cdot \left(4 {x}^{2} + 5\right)$

$= 2 x {e}^{{x}^{2}} \left(4 + 4 {x}^{2} + 5\right)$

$= 2 x {e}^{{x}^{2}} \left(4 {x}^{2} + 9\right)$