CHAPTER 7 - TECHNIQUES OF DEFFERENTIATION.pptx

TECHNIQUES OF DEFFERENTIATION

1. Chapter 7 TECHNIQUES OF DIFFERENTIATION

2. TECHNIQUES OF DEFFERNTIATION BASIC DERIVATIVE RULES Use various forms of derivative notation Use the Constant Rule, Power Rule, Constant Multiple Rule, and Sum and Difference Rule to find the derivative of a function T HE PRODUCT AND QUOTIENT RULES Use the Product Rule and Quotient Rule to find the derivative of a function THE CHAIN RULE Use the Chain Rule to find the derivative of a function EXPONENTIAL AND LOGARITHMIC RULES Use the Exponential Rule and Logarithmic Rule to find the derivative of a function IMPLICIT DIFFERENTIATION Use implicit differentiation to differentiate functions and nonfunctions

3. Basic Derivative Rules DERIVATIVE NOTATION The derivative of a function may be represented by any of the following: , read “f prime of x” , read “y prime” , read “dee why dee ex” or “the derivative of y with respect to x”  

4. Basic Derivative Rules CONSTANT RULE The derivative of a constant function is  

5. Finding the Derivative of a Constant Function Find the derivative of . Solution: 2. Find the derivative of Solution:   EXAMPLES:

6. Basic Derivative Rules THE POWER RULE The derivative of a function , where x is a variable and n is a nonzero constant, is  

7. Finding the Derivative of a Power Function 1. Find , given Solution: 2. Differentiate . Solution:   EXAMPLES:  

8. Basic Derivative Rules CONSTANT MULTIPLE RULE The derivative of a differentiable function with constant is given by  

9. Differentiate Solution:   EXAMPLE:

10. Basic Derivative Rules SUM AND DIFFERENCE RULE The derivative of a differentiable function is given by  

11. Differentiate Solution:   EXAMPLE:

12. Find given Solution:   EXAMPLE:

13. The Product and Quotient Rules PRODUCT RULE The derivative of a function is given by  

14. 1. Find the derivative of Solution:   EXAMPLE:    

15. We can check our work by multiplying out the factors of and differentiating the result using the Power Rule.   EXAMPLE:

16. 2. Differentiate 3. Differentiate   EXAMPLE:

17. The Product and Quotient Rules QUOTIENT RULE The derivative of a function is given by  

18. Find the derivative of Solution:   EXAMPLE:      

19. 2. Find the derivative of 3. Find the derivative of   EXAMPLE:

20. The Chain Rule GENERALIZED POWER RULE The derivative of the function is given by  

21. Differentiate Solution:   EXAMPLE:    

22. 2. Differentiate:   EXAMPLE:

23. Exponential and Logarithmic Rules Exponential Rule The derivative of the function is given by  

24. Differentiate Solution:   EXAMPLE:  

25. 2. Differentiate:   EXAMPLE:

26. Exponential and Logarithmic Rules Logarithmic Rule The derivative of the function is given by  

27. Differentiate Solution:   EXAMPLE:  

28. Exponential and Logarithmic Rules Generalized Logarithmic Rule The derivative of the function is given by  

29. Differentiate Solution:   EXAMPLE:  

30. 2. Differentiate:   EXAMPLE:

31. Implicit Differentiation Steps of Implicit Differentiation Differentiate both sides of the equation with respect to x 2. Algebraically, isolate the term  

32. Differentiate using implicit differentiation. Solution:   EXAMPLE: Product Rule

33.   EXAMPLE:

34. 2. F inding by Using Implicit Differentiation a. b. c.   EXAMPLE: