1. Chapter 4 Techniques of Differentiation Sections 4.1, 4.2, and 4.3

2. Techniques of Differentiation  The Product and Quotient Rules  The Chain Rule  Derivatives of Logarithmic and Exponential as Functions

3. Available Rules for Derivatives

4. Two More Rules The product rule The quotient rule If f (x) and g (x) are differentiable functions, then we have

5. The Product Rule - Example Derivative of first Derivative of Second

6. Derivative of numerator Derivative of denominator The Quotient Rule - Example

7. Calculation Thought Experiment Given an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient; and so on.

8. Example: To compute a value, first you would evaluate the parentheses then multiply the results, so this can be treated as a product. To compute a value, the last operation would be to subtract, so this can be treated as a difference. Calculation Thought Experiment Example:

9. The Chain Rule The derivative of a f (quantity) is the derivative of f evaluated at the quantity, times the derivative of the quantity. If f is a differentiable function of u and u is a differentiable function of x, then the composite f (u) is a differentiable function of x, and

10. Generalized Power Rule Example:

11. Generalized Power Rule Example:

12. Chain Rule in Differential Notation If y is a differentiable function of u and u is a differentiable function of x, then

13. Chain Rule Example Sub in for u

14. Logarithmic Functions Generalized Rule for Natural Logarithm Functions Derivative of the Natural Logarithm If u is a differentiable function, then

15. Find the derivative of Find an equation of the tangent line to the graph of Slope: Equation: Examples

16. Generalized Rule for Logarithm Functions. Derivative of a Logarithmic Function. If u is a differentiable function, then More Logarithmic Functions

17. Examples

18. Logarithms of Absolute Values

19. Examples

20. Exponential Functions Generalized Rule for the natural exponential function. Derivative of the natural exponential function. If u is a differentiable function, then

21. Find the derivative of Examples

22. Generalized Rule for general exponential functions. Derivative of general exponential functions. If u is a differentiable function, then Exponential Functions

23. Find the derivative of Exponential Functions