Question

How do you use the product Rule to find the derivative of `f(x) = (2x+1)(4-x^2)(1+x^2)`?

Answer 1

See the explanation.

Explanation:

`d/dx(gh) = g'h+gh'`

`d/dx(uvw) = d/dx(u(vw))`

`= u'(vw) + ud/dx(vw)`

`= u'(vw) + u(v'w + vw')`

`= u'vw + uv'w + uvw'`

So for `f(x) = (2x+1)(4-x^2)(1+x^2)`, we get

`f'(x) = (2)(4-x^2)(1+x^2)+(2x+1)(-2x)(1+x^2)+(2x+1)(4-x^2)(2x)`.

Rewrite to taste using algebra.