Question

How do you differentiate `f(x)=x(3x-9)^3` using the chain rule?

Answer 1

`f'(x) = 3(3x - 9 )^2(4x - 3 )`

Explanation:

The `color(blue) " Product rule " "will have to be used as well as the "`
`color(blue)" chain rule "`

There is a product of 2 functions here , x and `(3x - 9 )^3`

using the `color(blue) " Product rule "`

If f(x) = g(x).h(x) then f'(x) = g(x).h'(x) + h(x).g'(x)

using the`color(blue) " chain rule "`

`d/dx[f(g(x))] = f'(g(x)).g'(x)`

hence: `f'(x) = x d/dx(3x-9)^3 + (3x-9)^3 d/dx(x)`

`=x[3(3x-9)^2 d/dx(3x-9)] + (3x-9)^3 .1`

`=x[3(3x-9)^2 .3] + (3x-9)^3`

`=9x(3x-9)^2 + (3x-9)^3`

( can be simplified by 'taking out' common factor )

`= (3x-9)^2[9x +3x - 9 ] = (3x-9)^2 . 3(4x-3)`