# How do you differentiate g(x) = xe^(2x) using the product rule?

Aug 22, 2016

The result is (2x+1)e^(2x).

#### Explanation:

The rule says that $\frac{d}{\mathrm{dx}} f \left(x\right) \cdot g \left(x\right) = \mathrm{df} \frac{x}{\mathrm{dx}} \cdot g \left(x\right) + f \left(x\right) \cdot \mathrm{dg} \frac{x}{\mathrm{dx}}$.
In this case the two functions are $x$ and ${e}^{2 x}$.
Unfortunately the text call the product $g \left(x\right)$ and this can create some confusion with my description of the product rule.

$\frac{\mathrm{dx} {e}^{2 x}}{\mathrm{dx}} = \frac{\mathrm{dx}}{\mathrm{dx}} \cdot {e}^{2 x} + x \cdot \frac{d}{\mathrm{dx}} {e}^{2 x}$

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

$= \left(2 x + 1\right) {e}^{2 x}$.