Question

How do you differentiate `f(x)=4x(2x+3)^2` using the product rule?

Answer 1

`f'(x)=12(2x+3)(2x+1)`

Explanation:

The product rule states that for a function `f(x)=g(x)h(x)`, the derivative of the function is

`f'(x)=g'(x)h(x)+g(x)h'(x)`

Applying this to `f(x)=4x(2x+3)^2`, we see that

`f'(x)=color(red)(d/dx[4x])(2x+3)^2+4xcolor(blue)(d/dx[(2x+3)^2])`

We can find each of these derivatives separately and then plug them back in.

`color(red)(d/dx[4x]=4`

The following will require the chain rule.

`color(blue)(d/dx[(2x+3)^2])=2(2x+3)^1d/dx[2x+3]color(blue)(=4(2x+3)`

From these, we see that

`f'(x)=4(2x+3)^2+4x(4(2x+3))`

Factor out a `(2x+3)` term.

`f'(x)=(2x+3)(4(2x+3)+16x)`

Simplify.

`f'(x)=12(2x+3)(2x+1)`

This is also equal to

`f'(x)=48x^2+96x+36`