Question

What is the derivative of `f(x) = x^5 * (x^2-3)^`?

Answer 1

`7x^6-15x^4`

Explanation:

Using the product rule:

`d/dx (uw) = u'w + uw'`

This basically means in order to find the derivative of the function `(uw)`, you have to separate it into 2 separate functions, `u` and `w`. Then you find the derivative of `u` and `w` and multiply their derivatives `u'` and `w'` with the other function and add their products together.

Where
`color(white)(aaaaaaa)`(u)`color(white)(aaaaaa)`(w)
`color(white)(aaaaaaa)` `darr` `color(white)(aaaaa)` `darr`
`f(x) = (x^5)*(x^2-3)`

`u = x^5` `color(white)(aaaaaa)` `w=(x^2-3)`
`u' = 5x^4` `color(white)(aaaa)` `w'=2x`

`-------------------`

`=(5x^4)(x^2-3) + (2x)(x^5)`
`=5x^6-15x^4+2x^6`
`=7x^6-15x^4`