Question

Derivative of `x^5(3-6x^-9)`?

Answer 1

`f'(x)=54x^-5+5x^4(3-6x^-9)`

Explanation:

We have `f(x)=x^5(3-6x^-9)=g(x)h(x)`

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

`g(x)=x^5`
`g'(x)=5x^4`

`h(x)=3-6x^-9`
`h'(x)=-9*-6x^(-9-1)=54x^-10`

`f'(x)=x^5(54x^-10)+5x^4(3-6x^-9)`
`color(white)(f'(x))=54x^-5+5x^4(3-6x^-9)`

Answer 2

`f^'(x)=15x^4+24x^(-5)`

Explanation:

Set `y=x^5(3-6x^(-9))`

Multiply out the brackets

`y=3x^5-(6x^5)/x^9`

`y=3x^5-6/x^4`

`y=3x^5-6x^(-4)`

Or if you prefer

`f(x)=3x^5-6x^(-4)`

The next step is to differentiate each term.

Using the principle that if: `g(x)=ax^n -> g^'(x) =(axxn)x^(n-1)`

`f^'(x)=(3xx5)x^(5-1)-[6xx(-4)x^(-4-1)]`

`f^'(x)=15x^4+24x^(-5)`

They used to use `dy/dx` for `f^'(x)` You will still see this sometimes.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Check")`
Output from Maple:

`5x^4(3-6/x^9)+54/x^5`

`15x^4-30/x^5+54/x^5`

`15x^4-24/x^5`

`15x^4-24x^(-5) color(red)(larr" Confirmed")`