Question

How do you differentiate `g(z) = z^2cos(z^2-z)` using the product rule?

Answer 1

First, see that the two factors here are `z^2` and `cos(z^2-z)`.
Second, we'll need also the chain rule to derivate the second term.

Explanation:

• Product rule states that for `y=f(x)g(x)`, `(dy)/(dx)=(dy)/(du)(du)/(dx)`
• Chain rule states that `(dy)/(dx)=(dy)/(du)(du)/(dx)`

Here, we can rename `u=z^2-z` and now apply chain rule when we need to derivate the second term, following the product rule.

So:

`(dg(z))/(dz)=2zcos(z^2-z)+z^2u'sin(u)`

`(dg(z))/(dz)=2zcos(z^2-z)+z^2(2z-1)sin(z^2-z)`

`(dg(z))/(dz)=2z[cos(z^2-z)+(z^2-1)sin(z^2-z)]`