The Chain Rule is technically a method, but it's so important that it deserves its own section.
What is the Chain Rule?
Suppose that we have a function , and we want to differentiate it. How would we do it?
Remember that differentiation is the slope, which is
This isn't something we immediately know how to solve. What we do know how to solve is
What if was ? Then our numerator would be :
But, this isn't quite what we wanted - now the denominator doesn't match. No problem - we can multiply by the differentiation of :
That's it - that's the Chain Rule.
How long can the chain go?
Really, it's a chain - you choose the length. Rather, the problem chooses the length.
Remember during practice when you had to differentiate ? It was a pain for me to write the solution and for you to read it.
Now, we'll differentiate the same function in just a few easy steps! (No, that doesn't mean that the rest of the steps are hard).
Substitute from second step: