# How do you differentiate f(x)=(x+1)(x+2)(x+3) using the product rule?

Dec 31, 2015

You consider two of the factors as one, doing somewhat of a chain rule here.

#### Explanation:

In general terms, we can state that a product rule for three terms can be depicted as follows:

$\left(a b c\right) ' = \left(a b\right) ' c + \left(a b\right) c ' = \left(a ' b + a b '\right) c + \left(a b\right) c ' = \textcolor{g r e e n}{a ' b c + a b ' c + a b c '}$

Let's do it for your $f \left(x\right) = \left(x + 1\right) \left(x + 2\right) \left(x + 3\right)$:

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = \left(1\right) \left(x + 2\right) \left(x + 3\right) + \left(x + 1\right) \left(1\right) \left(x + 3\right) + \left(x + 1\right) \left(x + 2\right) \left(1\right)$

$\frac{\mathrm{df} \left(x\right)}{\mathrm{dx}} = \textcolor{g r e e n}{3 {x}^{2} + 12 x + 11}$