# How do you differentiate f(x)=(x-3lnx)(cosx+2x) using the product rule?

Mar 5, 2017

We will use the formula:

$y = f \left(x\right) \cdot g \left(x\right) \Rightarrow y ' = f ' \left(x\right) \cdot g \left(x\right) + f \left(x\right) \cdot g ' \left(x\right)$.

#### Explanation:

Detailed operations are as follows:

$f ' \left(x\right) = \left(x - 3 \ln x\right) ' \cdot \left(\cos x + 2 x\right) + \left(x - 3 \ln x\right) \cdot \left(\cos x + 2 x\right) '$

$f ' \left(x\right) = \left(1 - \frac{3}{x}\right) \cdot \left(\cos x + 2 x\right) + \left(x - 3 \ln x\right) \cdot \left(- \sin x + 2\right)$.

We could try to develop the products but the resulting expression can not be oversimplified and worth writing no more.