# Rules of Differentiation: The Chain Rule

### Erin Horst

Now that we have explored derivatives, we can now progress
to the rules of differentiation. These rules are simply formulas
that instruct the learner how to compute derivatives depending
on a given function.

The most complex (and useful) rule of differentiation is the
chain rule. The rule states, that if *f* and *g* are
both differentiable and *F=f(g(x))*, then* F* is differentiable
and *F'* is given by the product *F'(x)=f'(g(x))g'(x)*
(Stewart 1998).

To illustrate the chain rule, consider the following functions
and their respective derivatives. It is suggested that you compute
the derivative to verify the function of the derivative.

The follwing graph illustrates the function
and its derivative .

The following graph illustrates the function
and its derivative .

For additional practice with the chain rule, compute the derivatives
for the following functions:

The above problems were adapted from a Math Excel class instructed
by Joe Ediger and Erin Horst.

Stewart, J. (1998). *Calculus: Concepts and contexts*.
Pacific Grove, CA: Brooks/Cole Publishing Co.

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