The Chain Rule Lesson 4 3 Composition of

The Chain Rule Lesson 4. 3

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

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

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

Using the Calculator Given Define these functions on your calculator 5

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

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. 7

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 8

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 9

Chain Rule Example – given • Then and 10

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

Assignment Lesson 4. 3 Page 269 Exercises 1 – 65 EOO 12