Chapter Product Rule and Quotient Rule for Derivatives

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1 Chapter Product Rule and Quotient Rule for Derivatives Theorem 3.6: The Product Rule If f(x) and g(x) are differentiable at any x then

2 Example: The Product Rule. Find the derivatives:

3 Example: The Product Rule. Find the derivative :

4 Example: The Product Rule. Find the derivative :

5 Example: The Product Rule. Find the slope of the tangent line at x = 2 where

6 Product Rule (extended): The Product Rule Given that f(x), g(x), and h(x) are differentiable at any x then Example: The Product Rule. Find the derivative: when

7 Theorem 3.6: The Product Rule If f(x) and g(x) are differentiable at any x then Proof: Trick: Add and subtract the same value: -f(x)g(x+h) + f(x)g(x+h) Factor Separate Limit laws

8 Theorem 3.7 The Quotient Rule Given that f(x) and g(x) are differentiable at x then the derivative of exists provided Example: The Quotient Rule. Find the derivative:

9 Theorem 3.7 The Quotient Rule Given that f(x) and g(x) are differentiable at x then the derivative of exists provided Example: The Quotient Rule. Find the derivative:

10 Theorem 3.3 The Power Rule - revisited Let n be any integer (pos or neg) then Example: The Quotient Rule. Show that the derivative of is the same as the derivative of

11 Theorem 3.3 The Power Rule - revisited Let n be any integer (pos or neg) then Example: The Quotient Rule. Find the derivative:

12 Example: Find the equation of the tangent line at the point (3,2) of the graph of Question: Is it true that the derivative does not exist where y is undefined. (Not continuous implies not differentiable)

13 Example: Using the Product and Quotient Rule Find the derivative of Question: Does this derivative exist for all real numbers? What can you say about the continuity of f(x)?

14 Example: Using ANY rule...you decide Find the derivative of

15 Theorem 3.7 The Quotient Rule Given that f(x) and g(x) are differentiable at x then the derivative of exists provided Proof: The Quotient Rule.

16 Chapter Derivatives of Trigonometric Functions Recall Trig limits as. Use Squeeze Theorem Theorem 3.9 Trigonometric Limits

17 Theorem 3.9 Trigonometric Limits

18 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.

19 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.

20 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.

21 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.

22 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem. Use a change of variable.

23 Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem. Use a change of variable.

24 Exercises: Let for what values of 'a' is f(x) continuous?

25 Theorem 3.10 Derivative of Trigonometric Functions Proof:

26 Theorem 3.10 Derivative of Trigonometric Functions Graphs:

27 Theorem 3.10 Derivative of Trigonometric Functions Exercises:

28 Theorem 3.11 Derivative of Trigonometric Functions MEMORIZE!!! Exercise: Find the derivatives:

29 Theorem 3.11 Derivative of Trigonometric Functions MEMORIZE!!! Exercise: Find the derivatives:

30 Exercise: For which values does f(x) = x - sinx have a horizontal tangent line?

31 Exercise: For which values does f(x) = x - sinx have a slope of 1?

32

The Derivative of a Function Measuring Rates of Change of a function. Secant line. f(x) f(x 0 ) Average rate of change of with respect to over,

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