Let's not beat around the bush. Here you go:
Pretty amazing that the derivative of an ugly function like would be a pretty function like . It's kind of a Cinderella story for functions, but without the pumpkins, glass slipper and raging royal hormones.
Now, suppose that we want to differentiate . We have two options:
Option 1: Just do it by the chain rule .
Option 2: Use the fact that
In general, the chain rule says
So for instance,