Warm up Find the derivative of each function: 1. 2.

The Chain Rule, The Product Rule and the Quotient Rule Objective: To use the chain rule, the product rule and the quotient rule to find the derivative of more complicated functions.

Product Rule – used when functions are being multiplied Now cross multiply and place into the rule… g f Product Rule

2. Same derivative by expanding and using the Power Rule. 1. The Product Rule Find the derivative of.

Product Rule : Product Rule

The Quotient Rule This rule may look overwhelming with the functions but it is simply: The derivative of a quotient is the bottom times the derivative of the top minus the top times the derivative of the bottom over the bottom squared.

Example of the Quotient Rule: Find First we will find the derivative by using The Quotient Rule

Another way to do the same problem is to do the division first and then use the power rule. Again, notice there is more than one method you could use to find the derivative.

Example 2: Find the derivative of For this quotient doing the division first would require polynomial long division and is not going to eliminate the need to use the Quotient Rule. So you will want to just use the Quotient Rule.

The Chain Rule If h(x) = g(f(x)), then h’(x) = g’(f(x))f’(x). The Chain Rule deals with the idea of composite functions and it is helpful to think about an outside and an inside function when using The Chain Rule. In other words: The derivative when using the Chain Rule is the derivative of the outside leaving the inside unchanged times the derivative of the inside.

Using the Chain Rule, it will be helpful to identify the outside and inside before beginning. h(x) = (x+2) 4. The outside = ( ) 4 and the inside = x+2. Can you identify these? Now using the chain rule: Derivative = derivative of outside leaving inside * the derivative of the inside. h’(x) = (4(x+2) 3 )*(1) h’(x)= 4(x+2) 3 The Chain Rule

Another example: Find the derivative of It is helpful to identify the outside function and the inside function. In this example, the outside function is the cube, and the inside function is x The chain rule says take the derivative of the outside function leaving the inside function unchanged and then multiply by the derivative of the inside function. The derivative of the inside using the Power Rule The derivative of the outside leaving the inside unchanged

Find the derivative (solutions to follow)

Solutions

Sources Kelly, Greg. "The Chain Rule." Online Mathematics at the University of Houston. U of Houston, Web. 3 Apr Stewart. "The Chain Rule." Faculty at Bucks CCC. Bucks County Community College, n.d. Web. 3 Apr p...‎ Arnold, Julia, and Overman, Karen. "The Product and Quotient Rules." Tidewater Community College. Tidewater Community College, n.d. Web. 4 Apr Arnold, Julia, and Overman, Karen. "The Chain Rule." Tidewater Community College. Tidewater Community College, n.d. Web. 4 Apr