Derivatives of Composite Functions: The Chain Rule Section 3.7 (pages 107 – 110) Morgan Woods

Ladies and Gentlemen of the Math Emporium… In this presentation I hope to provide you, my fellow calculus scholars, with a clear understanding of how to work with composite functions. Section 3.7 deals with differentiation in composite functions. Due to my strong understanding of previous sections, which dealt with simpler forms of differentiation, I decided that my teaching skills would be put to best use in explaining this section

Didn’t you always want to know how to… Simplify a composite function Differentiate (find the derivative of) a composite function Learn the Chain Rule

Before we get into the good stuff, here’s some important review Note: this stuff will be important!! Derivative: –Instantaneous Rate of Change Differentiation: –Process of finding the Derivative How to differentiate Power Functions: IF.... f(x) = x n THEN.. f’(x) = nx n-1

So what is a composite function? Composite Function: f(x) = sin(x 2 ) Outside Function:sin Inside Function:x 2 Outside FunctionInside Function

The Chain Rule! The Chain rule actually has three parts, put for the sake of simplicity, let’s stick with this one. Outside Function, Inside Function Form: –Differentiate the outside function –Differentiate the inside function –multiply derivatives of the the inside function and outside functions Piece of cake, right?

Example Problem Problem: If f(x) = (x 3 + 7) 2, find f’(x). Process: f’(x) = 2(x 3 +7) 1 * 3x 2 Solution:f’(x) = 6x 2 (x 3 +7) Derivative Of Inside Function Derivative of outside function Multiply the two derivatives

Example Problem with Substitution Problem: If f(x) = (x 3 + 7) 2, find f’(x). Substitute: Let u be the inside function u=x 3 +7 and its derivative is u’=3x 2 So: f(x) = u 2 and f’(x)=2u * u’ Solution:f’(x) = 6x 2 (x 3 +7)

Composite Functions in the Real World!! Page 110: Question 25 A spherical balloon is being inflated with air. The volume of the sphere depends on the radius, and the radius depends on time. Thus, the volume is a composite function of time.

So in a Nutshell… A composite function is “composed” of two other functions: the inside function and the outside function The Chain Rule is used to find the derivative of a composite function…think of it as the secret recipe!! Composite functions can ACTUALLY be used in real world situations, like finding the volume of a balloon. Now, you can find the derivative of a composite function, and inversely, you can find the composite function if given the equation of the derivative function.

THIS IS THE THE END, MY FRIEND!