1. The Chain Rule
Lesson 4.3

2. Composition of Functions
Value fed to first function
Resulting value fed to second function 
End result taken from second function 

3. Composition of Functions
Notation for composition of functions:
Alternate notation:

4. Composition of Functions
Given two functions:
p(x) = 2x + 1
q(x) = x2 - 3
Then  p ( q(x) ) =
p (x2 - 3) =
2 (x2 - 3) + 1 =
2x2 - 5
Try determining  q ( p(x) ) 

5. Using the Calculator
Given
Define these functions on your calculator

6. Using the Calculator Now try the following compositions: g( f(7) )
f( g(3) )
g( f(2) )               
f( g(t) )
g( f(s) )
WHY ??

7. Decomposition of Functions
Someone once dug up Beethoven's tomb and found him at a table busily erasing stacks of papers with music writing on them.  They asked him ... "What are you doing down here in your grave?"  He responded, "I'm de-composing!!"
But, seriously folks ...
Consider the following function which could be a composition of two different functions.

8. Problem
Our example
We need a way to find the derivative of these kinds of functions
Without having to go through the trouble of raising the polynomial to the power
This is a function of composition – we need to "decompose" the function

9. Solution: The Chain Rule
Given y = f (u) and u = g (x)
That is y = f(u) = f ( g(x) )
Then
In words:
The derivative of y with respect to x is the derivative of y with respect to u times the derivative of u with respect to x

10. Chain Rule
Example – given
Then and

11. Note the alternative form of the chain rule definition
Try It Out
Consider the following functions of composition … find the derivatives
Note the alternative form of the chain rule definition

12. Assignment
Lesson 4.3
Page 269
Exercises 1 – 65 EOO