Friday, September 19, 2014

The Chain Rule

A friend of mine started taking a calculus class recently and I've been helping him with studying and homework. It had been a while for me but he had some questions about the chain rule. I thought the notation makes it kind of difficult. The chain rule is:
\[
f(g(x))' = f'(g(x))g'(x)
\]
But isn't $f(g(x))'$ the same thing as $f'(g(x))$? (No.) This is my explanation of how to apply the chain rule.

When not to use the chain rule

If you actually need to use other rules like the sum rule, product rule, or quotient rule.

When to use the chain rule

When you see an equation where you say something like, "I can find the derivative of part of that equation, and if that part were simply $x$ then I could solve the rest, too." You see a function within a function. It takes some practice. These are some pretty good examples.

How to solve the chain rule

The confusing part is that often the notation gives it as $f(g(x))' = f'(g(x))g'(x)$. What I prefer is longer but helps me.

You don't know the derivative to $h(x)$ but you can write it as
\[
h(x) = f(g(x))
\]
Write down $g(x)$ and $f(y)$ explicitly.
\[
f(y) =
\]
\[
g(x) =
\]
(I like to use $y$ instead of $x$ to try to help show the difference between $f$ and $g$.) Once those are written down, find $f'(y)$ and $g'(x)$. Write them out explicitly, also.
\[
f'(y) =
\]
\[
g'(x) =
\]
Now, wherever you still see a $y$ in $f'(y)$, replace it with $g(x)$ (the whole equation). That gives $f'(g(x))$. Now multiply $f'(g(x))$ by $g'(x)$. You're done! (Okay, you may not be done with algebra to simplify the result but you are done with the calculus part.)