# How do you differentiate f(x)=(1+x)(2-x) using the product rule?

Dec 1, 2015

$f ' \left(x\right) = 1 - 2 x$

#### Explanation:

According to the product rule, $\frac{d}{\mathrm{dx}} \left[f \left(x\right) g \left(x\right)\right] = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$.

We can say that:

$f ' \left(x\right) = \left(2 - x\right) \frac{d}{\mathrm{dx}} \left[1 + x\right] + \left(1 + x\right) \frac{d}{\mathrm{dx}} \left[2 - x\right]$

Note that:

$\frac{d}{\mathrm{dx}} \left[1 + x\right] = 1$

$\frac{d}{\mathrm{dx}} \left[2 - x\right] = - 1$

Thus:

$f ' \left(x\right) = \left(2 - x\right) \left(1\right) + \left(1 + x\right) \left(- 1\right)$

$f ' \left(x\right) = 2 - x - 1 - x$

$f ' \left(x\right) = 1 - 2 x$