2.4 The Chain Rule

The Chain Rule If y = f(u) and u = g(x) (both appropriately differentiable), then the composite function y = f(g(x)) is differentiable.

METHOD 1 – Keep f and g separate

Example: If f and g are defined separately:

#4

At a given point:

Discussion pg. 140 “where to put the value in for x”

#16 “where to put the value in for x”

METHOD 2 – f(g(x)) is already composed

This occurs MOST often.

At this point, most of our problems use the special case:

General Power Rule

#26 #30

Use in conjunction with product or quotient rule

Recognize “which rule to use first”

For these four examples, we won’t find the derivative, but determine the “rule sequence”.

Example

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Write Equation of tangent line

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Find values where the tangent is horizontal; i.e. f’(x) = 0

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Applications

#66 MANUFACTURING COST

At a certain factory the total cost of manufacturing q units is dollars. It has been determined that approximately units are manufactured during the first t hours of a production run. Compute the rate at which the total manufacturing cost is changing with respect to time 1 hour after production commences.

**Note – Read pg 153, #67 together a) asking for b) Asking for