The Chain Rule



The Chain Rule

Composite Functions

If f(x) = 2x2 – 7x and g(x) = 3x – 5 than:

f(g(x)) =

How could (4 + 2x)2 be written as a composite function?

How could -3(6 – 5x)3 be expressed as a composite function?

The Chain Rule will be used to find derivatives of composite functions. There are two definitions for the chain rule, both mean the same thing:

If y is a function of u (y = f(u)) and u is a function of x (u = g(x)), than y = f(u) = f(g(x)), and :

[pic]

OR

If y = f(g(x)), than [pic]

Example 1:

Find [pic] if [pic]

Example 2: (Product and Chain Rules)

If [pic] find [pic]

Example 3: (Quotient and Chain Rules)

If [pic] , find [pic]

Example 4:

Find [pic]when [pic]

Example 5:

Find [pic]when [pic]

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