Question

How do you use the product Rule to find the derivative of `y (x) = (x^2+1)(x^3+1)`?

Answer 1

I found: `y'(x)=5x^4+3x^2+2x`

Explanation:

The product rule allows you to derive a function formed by the product of two functions as `f(x)=g(x)*h(x)` to get:
`f'(x)=g'(x)h(x)+g(x)h'(x)`
In your case you have (in red are the derivatives):
`y'(x)=color(red)(2x)(x^3+1)+(x^2+1)color(red)(3x^2)=`
`=2x^4+2x+3x^4+3x^2=5x^4+3x^2+2x`

Answer 2

You can always multiply it out if you don't want to use the product rule.

Product Rule:
`d/(dx)[f(x)g(x)] = f(x)(dg(x))/(dx) + g(x)(df(x))/(dx)`

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

`= (3x^4 + 3x^2) + (2x^4+2x)`

`= color(blue)(5x^4 + 3x^2 + 2x)`

You can also do this with the Power Rule:

`h(x) = (x^2+1)(x^3+1)`

`h(x) = x^5 + x^2 + x^3 + 1`

`d/(dx)[h(x)]`

`=> 5x^4 + 2x + 3x^2`

`= color(blue)(5x^4 + 3x^2 + 2x)`