Question

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

Answer 1

`f'(x) = -2(2x-3)^2(5x^2+3x-12)`

Explanation:

We have:

`f(x )= (4-x^2)(2x-3)^3`

So using the product rule we have:

`f'(x )= {4-x^2}{ d/dx (2x-3)^3} + {d/dx (4-x^2)}{(2x-3)^3}`

So using the chain rule we have:

`f'(x) = {4-x^2}{ 3(2x-3)^2(2)} + {-2x}{(2x-3)^3}`
`" " = 6(4-x^2)(2x-3)^2 -2x(2x-3)^3`
`" " = 2(2x-3)^2{3(4-x^2) -x(2x-3)}`
`" " = 2(2x-3)^2(12-3x^2 -2x^2+3x)`
`" " = 2(2x-3)^2(-12-5x^2-3x)`
`" " = -2(2x-3)^2(5x^2+3x-12)`