# How do you differentiate f(x)=xe^(3x) using the product rule?

Jan 10, 2017

$f ' \left(x\right) = {e}^{3 x} \left(1 + 3 x\right)$

#### Explanation:

The product rule for differentiation is:

$f \left(x\right) = u v ,$ where $u$ & $v$ are both functions of $x$

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

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

$u = x \implies u ' = 1$

$v = {e}^{3 x} \implies v ' = 3 {e}^{3 x}$

$\therefore f ' \left(x\right) = {e}^{3 x} \times 1 + x \times 3 {e}^{3 x}$

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

$f ' \left(x\right) = {e}^{3 x} \left(1 + 3 x\right)$