Question

Question #16171

Answer 1

`f'(x) = 2(2x-1)^2(x^2+3) (7x^2-2x+9)`

Explanation:

We can use a combination of the chain rule and the product rule:

We have:

`f(x) = (2x-1)^3(x^2+3)^2`

We can use the chain rule on each individual function. Let

`{ (u=(2x-1)^3), (v=(x^2+3)^2) :} => { ((du)/dx=3(2x-1)^2(2),=6(2x-1)^2), ((dv)/dx=2(x^2+3)(2x),=4x(x^2+3)) :}`

And then:

`f(x) = uv`

And by the product rule we have:

`f'(x) = (u)((dv)/dx) + ((du)/dx)(v)`
`" " = (2x-1)^3 4x(x^2+3) + 6(2x-1)^2(x^2+3)^2)`
`" " = (2x-1)^2(x^2+3) {4x(2x-1)+6(x^2+3)}`
`" " = (2x-1)^2(x^2+3) (8x^2-4x+6x^2+18)`
`" " = (2x-1)^2(x^2+3) (14x^2-4x+18)`
`" " = 2(2x-1)^2(x^2+3) (7x^2-2x+9)`