Question

How do you differentiate `f(x)= (x^2+1) (e^x+2)^2 (x^-3-3)^3` using the product rule?

Answer 1

Use the product rule twice (along with the power and chain rules).

Explanation:

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

`= u'(vw)+u[d/dx(vw)]`

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

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

Apply either the formula or the process to this question with

`u = x^2+1` `" "` `" "` `" "` `" "` So `u' = 2x`
`v = (e^x+2)^2` `" "` `" "` `" "` So `v'=2e^x(e^x+2)`
`w = (x^-3+3)^3` `" "` `" "` So `w' = -9x^-4(x^-3+3)^2`