Chain Rule

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 f(g(x))f(g(x)), and we want to differentiate it. How would we do it?

Remember that differentiation is the slope, which is

Change in f(g(x))Change in x=df(g(x))dx.\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

d f(n)dn\frac{d~ f(n)}{d n}

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

d f(g(x))d 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)g(x):

d f(g(x))d g(x)d g(x)dx=df(g(x))dx.\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.

How long can the chain go?

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

For example,

d d(e(f(g(x))))d e(f(g(x)))d e(f(g(x)))d f(g(x))d f(g(x))d g(x)d g(x)dx=d d(e(f(g(x))))dx.\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 lnx\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=lnxy=\ln x
x=eyx = e^y
dxdx=deydx\frac{dx}{dx} = \frac{d e^y}{dx}
dxdx=deydydydx\frac{dx}{dx} = \frac{d e^y}{dy}\cdot \frac{dy}{dx}
1=eydydx1 = e^y \frac{dy}{dx}
1ey=dydx\frac{1}{e^y} = \frac{dy}{dx}

Substitute from second step:

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