Chapters

Chain Rule

The Chain Rule is technically a method, but it's so important that it deserves its own section.

Suppose that we have a function $f(g(x))$ , and we want to differentiate it. How would we do it?

Remember that differentiation is the slope, which is

\frac{\text{Change in }f(g(x))}{\text{Change in } x}=\frac{d f(g(x))}{d x}.

This isn't something we immediately know how to solve. What we do know how to solve is

\frac{d~ f(n)}{d n}

What if $n$ was $g(x)$ ? Then our numerator would be $f(g(x))$ :

\frac{d~f(g(x))}{d~g(x)}.

But, this isn't quite what we wanted - now the denominator doesn't match. No problem - we can multiply by the differentiation of $g(x)$ :

\frac{d~f(g(x))}{\cancel{d~g(x)}}\cdot \frac{\cancel{d~g(x)}}{d x} =\frac{d f(g(x))}{d x} .

That's it - that's the Chain Rule.

Really, it's a chain - you choose the length. Rather, the problem chooses the length.

For example,

\frac{d~d(e(f(g(x))))}{d~e(f(g(x)))}\cdot\frac{d~e(f(g(x)))}{d~f(g(x))}\cdot\frac{d~f(g(x))}{d~g(x)}\cdot \frac{d~g(x)}{d x} =\frac{d~d(e(f(g(x))))}{d x}.

Remember during practice when you had to differentiate $\ln x$ ? 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).

y=\ln x

x = e^y

\frac{dx}{dx} = \frac{d e^y}{dx}

\frac{dx}{dx} = \frac{d e^y}{dy}\cdot \frac{dy}{dx}

1 = e^y \frac{dy}{dx}

\frac{1}{e^y} = \frac{dy}{dx}

Substitute from second step:

\boxed{\frac{1}{x} = \frac{dy}{dx}}