# How do you differentiate f(x)= 2xlnx using the product rule?

Nov 10, 2015

$\frac{d}{\mathrm{dx}} 2 x \ln \left(x\right) = 2 \left(\ln \left(x\right) + 1\right)$

#### Explanation:

The product rule states
$\frac{d}{\mathrm{dx}} f \left(x\right) g \left(x\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

In this case, we have $f \left(x\right) = 2 x$ and $g \left(x\right) = \ln \left(x\right)$
$\frac{d}{\mathrm{dx}} 2 x = 2$
$\frac{d}{\mathrm{dx}} \ln \left(x\right) = \frac{1}{x}$

So

$\frac{d}{\mathrm{dx}} 2 x \ln \left(x\right) = 2 \cdot \ln \left(x\right) + 2 x \cdot \frac{1}{x} = 2 \left(\ln \left(x\right) + 1\right)$