Chapter 3.3 - 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
Example: The Product Rule. Find the derivatives:
Example: The Product Rule. Find the derivative :
Example: The Product Rule. Find the derivative :
Example: The Product Rule. Find the slope of the tangent line at x = 2 where
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
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
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:
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:
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
Theorem 3.3 The Power Rule - revisited Let n be any integer (pos or neg) then Example: The Quotient Rule. Find the derivative:
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)
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)?
Example: Using ANY rule...you decide Find the derivative of
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.
Chapter 3.4 - Derivatives of Trigonometric Functions Recall Trig limits as. Use Squeeze Theorem Theorem 3.9 Trigonometric Limits
Theorem 3.9 Trigonometric Limits
Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.
Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.
Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.
Theorem 3.9 Trigonometric Limits Exercises: How to use this theorem. Rewrite so that it has the same structure as the theorem.
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.
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.
Exercises: Let for what values of 'a' is f(x) continuous?
Theorem 3.10 Derivative of Trigonometric Functions Proof:
Theorem 3.10 Derivative of Trigonometric Functions Graphs:
Theorem 3.10 Derivative of Trigonometric Functions Exercises:
Theorem 3.11 Derivative of Trigonometric Functions MEMORIZE!!! Exercise: Find the derivatives:
Theorem 3.11 Derivative of Trigonometric Functions MEMORIZE!!! Exercise: Find the derivatives:
Exercise: For which values does f(x) = x - sinx have a horizontal tangent line?
Exercise: For which values does f(x) = x - sinx have a slope of 1?