Question

How do you differentiate `y=(x^4+3)(-4x^5+5x^4+5)` using the product rule?

Answer 1

`y'=-4x^3(9x^5-10x^4+15x-20)`

Explanation:

Denote `f(x)=x^4+3` and `g(x)=-4x^5+5x^4+5` so that `y` is of the form `f.g`.

The derivative of `y` with respect to `x` according the product rule is then `f'.g+f.g'`:

`y' =f'.g+f.g'`
`y'=(4x^3)(-4x^5+5x^4+5)+(x^4+3)(-20x^4+20x^3)`
`y'=-16x^8+20x^7+20x^3-20x^8+20x^7-60x^4+60x^3`
`y'=-36x^8+40x^7-60x^4+80x^3`
`y'=-4x^3(9x^5-10x^4+15x-20)`