Download Product Rule and Quotient Rule Lesson Objectives

Lesson Objectives
• 1. Use first principles to develop the product
and quotient rules
• 2. Use graphic differentiation to verify the
product and quotient rules
• 3. Apply the product and quotient rules to
an analysis of functions
• 4. Apply the product and quotient rules to
real world problems
B.4.2 - Product Rule and
Quotient Rule
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(A) Review
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Explore
• The power rule tells us how to find the
derivative of any power function y = xn
which works for any real value of n
• Let’s define f(x) = x3 and g(x) as x6
• Then, determine d/dx (f (x) + g(x)) and summarize with a
new derivative expression
• Then determine d/dx (f(x) - g(x)) and summarize with a
new derivative expression
• The derivative of a sum/difference is simply
the sum/difference of the derivatives i.e.
(f + g)` = f ` + g`
• Use graphs to verify that your final answer is correct
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• Now, determine d/dx (f(x)*g(x)) and summarize with a
new derivative expression. Use graphs to verify that your
final answer is correct
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Explore
Explore
• Conclusion to our exploration ==> the
derivative of a product is NOT equal to the
product of the respective deriavtives!
• So how do we determine the derivative of a
product of two functions?
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• So why does (fg)` = f ` x g` not work?
• Let’s go back to limits and basic principles to find
what the differentiation technique should be if we
wish to find a derivative of a product
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K ( x + h) # K ( x )
h
f ( x + h) " g ( x + h) # f ( x ) " g ( x )
K !( x) = lim
h *0
h
f ( x + h) " g ( x + h) # f ( x + h) g ( x ) + f ( x + h) g ( x ) # f ( x ) " g ( x )
K !( x) = lim
h *0
h
g ( x + h) # g ( x )
f ( x + h) # f ( x ) &
)
K !( x) = lim ' f ( x + h) "
+ g ( x) "
$%
h *0
h
h
(
g ( x + h) # g ( x )
f ( x + h) # f ( x )
K !( x) = lim f ( x + h) " lim
+ lim g ( x) " lim
h *0
h *0
h *0
h *0
h
h
K !( x) = f ( x) " g !( x) + g ( x) " f !( x)
K !( x) = lim
h *0
• Let K(x) = [f(x)] x [g(x)]
• Then K `(x) = lim h→0 1/h[K(x + h) – K(x)]
• And K `(x) = lim h→0 1/h [ f(x+h)g(x+h) – f(x)g(x)]
which gets simplified as on the next slide:
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(B) Product Rule - Derivation
(B) Product Rule - Derivation
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(C) Product Rule - Examples
(C) Product Rule - Examples
• Ex 4. Find the equation of the tangent to the
graph of the function m(x) = (5x2 - 2)(x 4x2) at x = 1
• ex 1. Find the derivative of f(x) = 3x4(5x3 + 5x - 7)
• ex 2. Find the instantaneous rate of change at x = 1 of f(x)
= (x4 - 4x3 – 2x2 + 5x + 2)2
• Ex 5. If f(x) = x0.5*g(x), where g(4) = 2 and
g’(4) = 3, find f’(4)
• ex 3. Find the equation of the tangent to the function f(x) =
(2x + 4)(3x3 – 3x2 + x - 2) at (1,-6)
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(E) Quotient Rule Derivation
(D) Derivatives of Rational
Functions – The Quotient Rule
• Since the derivative of a product does not equal the
product of the derivatives, what about a quotient?
f ( x)
g ( x)
f ( x) = H ( x) # g ( x)
f !( x) = H !( x) # g ( x) + H ( x) # g !( x)
f !( x) " H ( x) # g !( x)
H !( x) =
g ( x)
f ( x)
f !( x) "
# g !( x)
g ( x)
H !( x) =
g ( x)
f !( x) g ( x) " f ( x) g !( x)
H !( x) =
[ g ( x)]2
H ( x) =
• Since quotients are in one sense nothing more than
products of a function and a reciprocal  we would guess
that the derivative of a quotient is not equal to the quotient
of the derivatives
• First, set up a division
and then rearrange the
division to produce a
multiplication so that
we can apply the
product rule developed
earlier
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• Would the derivative of a quotient equal the quotient of the
derivatives?
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(F) Examples Using the
Quotient Law
(G) Internet Links
• Differentiate each of the
following rational functions
• ex 1.
• ex 2.
• ex 3.
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• Calculus I (Math 2413) - Derivatives Product and Quotient Rule
2x + 5
3x ! 1
x3 ! 3
g ( x) =
1+ 4x2
x2
h( x ) =
( x ! 2)( x 4 ! x 2 )
f ( x) =
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• Visual Calculus - Calculus@UTK 3.2
• solving derivatives step-by-step from
Calc101
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(H) Day #2 of Lesson
Opening Exercise - Lesson Day 2
• Show your work in answering the following
multiple choice question
• FAST FIVE Table Quiz: You have 10 minutes
• Make a 2 column worksheet. In the first column,
briefly state what needs to be done and why and in
the second column, do it!
f (x) =
1
• For what intervals is the function
x2 +1
concave down? (HINT: Expand denominator
when taking d’)
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!
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(H) Group Q. #1
(H) Group Q. #1 - HINTS
• Again, make a 2 column worksheet. In the first
column, briefly state what needs to be done and
why and in the second column, simply do what
you said needs to be done! Sol’n to be submitted.
• Q1. Find the intervals of increase/decrease and
intervals of concavity for the given function. Then
sketch the function based on your intervals.
y=
• Determine the domain of f(x)
• Determine the vertical asymptotes and the
behaviour along the VA using limits
• Determine the horizontal asymptotes using
limits
• Determine the x- and y-intercepts
3x + 7
2x + 5
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(I) Group Q. #2
(J) Examples - Economics
• Again, make a 2 column worksheet. In the first
column, briefly state what needs to be done and
why and in the second column, simply do what
you said needs to be done! Sol’n to be submitted.
• Recall this example from Lesson 4.1 ==> Suppose that the
total cost in hundreds of dollars of producing x barrels of
oil is given by the function C(x) = 4x2 + 100x + 500.
Determine the following.
•
•
•
•
(a) the cost of producing 5000 barrels of oil
(b) the cost of producing 5001 barrels of oil
(c) the cost of producing the 5001st barrel of oil
(d) C `(5000) = the marginal cost at a production level of
5000 barrels of oil. Interpret.
• (e) The production level that minimizes the average cost
(where AC(x) = C(x)/x))
• Q2. Find the intervals of increase/decrease and
intervals of concavity for the given function.
Sketch f(x) as well.
2
f (x) =
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x"3
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(J) Examples - Economics
(H) Homework
• Text, S4.2, p234
• So it will be expected of you to calculate
algebraically and using the TI-89 and then
interpret a marginal cost, a marginal
revenue, a marginal profit, a marginal
average cost, etc…
• (1) Algebra: Q1-28 odds (hopefully done after our first
lesson )
• (2) Word Problems: Q36-47
• (3) Word problems: Q50,51,53,54,55 from pg 225
• So let’s try some in class together. See
handout.
• I will be MARKING (2) and (3) as part of your HW
mark!! I WILL BE considering the quality and
completeness of your solutions! There will also be a
HW quiz based on your solutions.
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ECO
• Find 2 alternate derivations of the product
rule, present and explain them to me
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