1. The Chain Rule
Section 2.4

2. After this lesson, you should be able to:
Find the derivative of a composite function using the Chain Rule.
Find the derivative of a function using the General Power Rule.
Simplify the derivative using algebra.
Find the derivative of a trigonometric function using the Chain Rule.

3. The Chain Rule
(deals with composition of functions)
If f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f(g(x)) is a differentiable function of x and
or,

4. Extended Power Rule
(power rule)
(extended (or general) power rule)

5. Example Given: y = (6x3 – 4x + 7)3
Then u(x) = 6x3 – 4x + 7 and f(u) = u3
Thus f’(x) = 3(6x3 – 4x + 7)2(18x2 – 4)

6. Try
Which is the, the “inner” function?
Which is the , the “outer” function?
What is the answer ??

7. Try
Which is the, the “inner” function?=
Which is the , the “outer” function?
What is the answer ??

8. Try
Which is the, the “inner” function? u =
Which is the , the “outer” function? =
What is the answer ??

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32. More Trig Derivatives…

33. Find the derivative.

34. Find the derivative.

35. Book Example
Example: Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.
pt: (2, 2)

36. Example: Given the function and a point:
a) Find an equation of the tangent line to the graph of f at a given point.
b) Use your calculator to graph the function and its tangent line
c) Use the derivative feature on the calculator to confirm your results.
pt: (1, 4)