1 The Product and Quotient Rules and Higher Order Derivatives Section 2.3

2 After this lesson, you should be able to: Find the derivative of a function using the Product Rule Find the derivative of a function using the Quotient Rule Find the derivative of a trig function Find a higher-order derivative of a function

3 The Product Rule Example: Find dy/dx : Let f and g be differentiable functions and let and

4 Product Rule Example Example: Find dy/dx :

5 Product Rule Example Example: Find f ’(x):

6 Product Rule Example Example: Find y’ :

7 The Quotient Rule Let f and g be differentiable functions such that and let and

8 Quotient Rule Example 1) Find y ’:

9 Quotient Rule Example 2) Find f ’(x):

10 3) Find Quotient Rule Example

11 4) Find Quotient Rule Example

12 5) Find Quotient Rule Example

13 The Derivative of Tangent The derivatives of the four remaining trig functions can be found using the derivatives of sine and/or cosine and the Quotient Rule. For example,

14 The Derivative of Cosecant

15 The Derivatives of the Six Basic Trig Functions Now for fun, you can use the Quotient Rule to find the derivatives of the two remaining trig functions! Here's a summary of the derivatives of the six trigonometric functions:

16 Trig Derivative Example Find the derivative of

17 Example: Write the equation of the tangent line at Equation of the Tangent Line

18 Higher Order Derivatives y =f(x) Ex:

19 Higher Order Derivatives of sin x

20 Higher Order Derivatives Example 1) Find the second derivative of

21 Higher Order Derivatives Example 2) Given a) Find b) Determine the point(s) at which the function has a horizontal tangent.

22 Homework Section 2.3 page 126 #1-23 odd, 29, odd, 51, 63, 67, 73, 99, 101, 116