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Study Guide

Combining the derivatives of basic functions with the chain rule gives us a lot of patterns that let us take derivatives of functions that seem complicated.

Let *h*(*x*) = *e*^{cos x}. If we think of this as *h*(*x*) = *f*(*g*(*x*)) where *f*(□) = *e*^{□} and *g*(*x*) = cos *x*,

the chain rule tells us that

*h*'(*x*) =* f* ' (*g*(*x*)) · *g ' *(*x*)

=* e*^{cos x} · (cos *x*)'

= *e*^{cos x} · (-sin *x*)

If instead we use Leibniz notation, we have *z* = *e ^{y}* where

.

Thankfully, we find the same thing either way: the derivative is the original function e

We can state this formally as

(*e ^{u}*)' =

assuming that the prime notation means "take the derivative with respect to *x*." Using Leibniz notation, we would say

There's also a less formal way that might make more sense:

if *h*(*x*) = e^{{□}}, then the derivative of *h* is

*h ' *(*x*) =* e*^{{□}} × (□)'

Similarly, the chain rule tells us that

and so on and so forth. These are good patterns to know, because then we can find derivatives without having to think much about the chain rule.

### Thinking Backwards

The chain rule patterns also help us to think backwards, which will be useful for something called "integration by substitution".

We'll get to that later, but what we mean by this is that given a derivative, the chain rule patterns we've covered may help us find the original function. More precisely, given

*f*' (*x*) we can venture a guess as to what*f*(*x*) might be.We'll explore this further through the examples and exercises.

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