Chapt3.3and3.4.notebook
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
October 03, 2014
Chapt3.3and3.4.notebook
Example: The Product Rule.
Find the derivatives:
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Chapt3.3and3.4.notebook
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Example: The Product Rule.
Find the derivative
:
Chapt3.3and3.4.notebook
October 03, 2014
Example: The Product Rule.
Find the derivative
:
Chapt3.3and3.4.notebook
Example: The Product Rule.
Find the slope of the tangent line at x = 2 where
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Chapt3.3and3.4.notebook
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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
Chapt3.3and3.4.notebook
October 03, 2014
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
Chapt3.3and3.4.notebook
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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:
Chapt3.3and3.4.notebook
October 03, 2014
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:
Chapt3.3and3.4.notebook
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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
Chapt3.3and3.4.notebook
Theorem 3.3 The Power Rule - revisited
Let n be any integer (pos or neg) then
Example: The Quotient Rule.
Find the derivative:
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Chapt3.3and3.4.notebook
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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)
Chapt3.3and3.4.notebook
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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)?
Chapt3.3and3.4.notebook
Example: Using ANY rule...you decide
Find the derivative of
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Chapt3.3and3.4.notebook
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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.
Chapt3.3and3.4.notebook
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Chapter 3.4 - Derivatives of
Trigonometric Functions
Recall Trig limits as
. Use Squeeze Theorem
Theorem 3.9 Trigonometric Limits
Chapt3.3and3.4.notebook
Theorem 3.9 Trigonometric Limits
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Chapt3.3and3.4.notebook
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Theorem 3.9 Trigonometric Limits
Exercises: How to use this theorem. Rewrite so that it has the
same structure as the theorem .
Chapt3.3and3.4.notebook
October 03, 2014
Theorem 3.9 Trigonometric Limits
Exercises: How to use this theorem. Rewrite so that it has the
same structure as the theorem .
Chapt3.3and3.4.notebook
October 03, 2014
Theorem 3.9 Trigonometric Limits
Exercises: How to use this theorem. Rewrite so that it has the
same structure as the theorem .
Chapt3.3and3.4.notebook
October 03, 2014
Theorem 3.9 Trigonometric Limits
Exercises: How to use this theorem. Rewrite so that it has the
same structure as the theorem .
Chapt3.3and3.4.notebook
October 03, 2014
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.
Chapt3.3and3.4.notebook
October 03, 2014
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.
Chapt3.3and3.4.notebook
Exercises: Let
of 'a' is f(x) continuous?
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for what values
Chapt3.3and3.4.notebook
Theorem 3.10 Derivative of Trigonometric Functions
Proof:
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Chapt3.3and3.4.notebook
Theorem 3.10 Derivative of Trigonometric Functions
Graphs:
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Chapt3.3and3.4.notebook
Theorem 3.10 Derivative of Trigonometric Functions
Exercises:
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Chapt3.3and3.4.notebook
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Theorem 3.11 Derivative of Trigonometric Functions
MEMORIZE!!!
Exercise: Find the derivatives:
Chapt3.3and3.4.notebook
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Theorem 3.11 Derivative of Trigonometric Functions
MEMORIZE!!!
Exercise: Find the derivatives:
Chapt3.3and3.4.notebook
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Exercise: For which values does f(x) = x - sinx have a horizontal
tangent line?
Chapt3.3and3.4.notebook
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Exercise: For which values does f(x) = x - sinx have a slope of 1?
Chapt3.3and3.4.notebook
October 03, 2014