1. 2.4 The Chain Rule
If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given by the product
In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then

2. Example:
A faster way to write the solution:
Differentiate the outer function...
…then the inner function

3. Another example:
It looks like we need to use the chain rule again!
derivative of the
outer function
derivative of the
inner function

4. Another example:
The chain rule can be used more than once.
(That’s what makes the “chain” in the “chain rule”!)

5. The Power Rule combined with the Chain Rule
If n is any real number and u=g(x) is differentiable, then
Alternatively,
Example: