The chain rule is one of those tricky topics that you have to practice a lot to get right. Basically it says that to take the derivative of a composite function 
, you take the derivate of 
, take the derivative of 
, and then multiply them in a special way: 
. It’s easiest to see with some examples.
What if we’re asked to find 
if 
. We might try multiplying 
times itself five times, but that’s tedious and boring. Instead, let’s think of 
as the composition of the functions 
and 
, so that 
. (Check to make sure you understand this last equation!)
If we write 
like that, then the chain rule tells us that 
. So we have to find 
and 
. But both of these are easy! We know that the derivative of 
is 
, and the derivative of is 
. So, 
. Cool!
For a harder one, how about finding the derivative of 
. Here we have the composition of two functions 
and 
. The functions are the same! Weird. In any case, the derivative of 
is 
, so the derivative of is 
.
With practice, you can find the derivatives of complicated functions without naming and . The way to do this is to differentiate the outer function, and then multiply by the derivative of the next inner function.
