10.6 The chain rule
The chain rule is used to find the derivatives of "functions of a function". Mathematically, these are written as
and it’s possible to find this just by known the derivatives of and .
The proof is a little involved, so for the moment you can find it at http://kruel.co/math/chainrule.pdf.
The result we’re after is that
Find the derivative of
What is the first thing to do when approaching a question (well after having considered that the chain rule might be relevant)? We must find the nested functions. I find it helpful to think about the parts I find hard to differentiate. For example, in our example, I know how to differentiate , and also but not
Therefore, it is not unreasonable to create a variable and try to use the chain rule. We can write
This can be a little confusing, because no longer seems to appear in the function, but in reality is implicitly a function of (i.e. we know that depends on ). Applying the chain rule we know that
We now know how to differentiate all the individual pieces. First we know that
Then we can also differentiate with respect to , which is just
Therefore we get that overall