The product rule is generally better because:
- It is more systematic and avoids mistakes from expanding products
- It works for any differentiable functions u and v, not just polynomials
- It provides insight into the structure of the derivative that direct computation does not
So in this example, using the product rule is preferable to direct multiplication.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
Lesson 24: Areas, Distances, the Integral (Section 021 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
Continuity says that the limit of a function at a point equals the value of the function at that point, or, that small changes in the input give only small changes in output. This has important implications, such as the Intermediate Value Theorem.
In this video we learn how to solve limits that involve trigonometric functions. It is all based on using the fundamental trigonometric limit, which is proved using the squeeze theorem.
For more lessons: http://www.intuitive-calculus.com/solving-limits.html
Watch video: http://www.youtube.com/watch?v=1RqXMJWcRIA
Lesson 24: Areas, Distances, the Integral (Section 021 slides)Matthew Leingang
We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
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Many problems in nature are expressible in terms of a certain differential equation that has a solution in terms of exponential functions. We look at the equation in general and some fun applications, including radioactivity, cooling, and interest.
Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.
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The line tangent to a curve is also the line which best "fits" the curve near that point. So derivatives can be used for approximating complicated functions with simple linear ones. Differentials are another set of notation for the same problem.
The derivative of a composition of functions is the product of the derivatives of those functions. This rule is important because compositions are so powerful.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
Using implicit differentiation we can treat relations which are not quite functions like they were functions. In particular, we can find the slopes of lines tangent to curves which are not graphs of functions.
There's an obvious guess for the rule about the derivative of a product—which happens to be wrong. We explain the correct product rule and why it has to be true, along with the quotient rule. We show how to compute derivatives of additional trigonometric functions and new proofs of the Power Rule.
First principle, power rule, derivative of constant term, product rule, quotient rule, chain rule, derivatives of trigonometric functions and their inverses, derivatives of exponential functions and natural logarithmic functions, implicit differentiation, parametric differentiation, L'Hopital's rule
Integration by substitution is the chain rule in reverse. NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
Integration by substitution is the chain rule in reverse.
NOTE: the final location is section specific. Section 1 (morning) is in SILV 703, Section 11 (afternoon) is in CANT 200
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We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
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We can define the area of a curved region by a process similar to that by which we determined the slope of a curve: approximation by what we know and a limit.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
At times it is useful to consider a function whose derivative is a given function. We look at the general idea of reversing the differentiation process and its applications to rectilinear motion.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
Uncountably many problems in life and nature can be expressed in terms of an optimization principle. We look at the process and find a few good examples.
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See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
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- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
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Length: 30 minutes
Session Overview
-------------------------------------------
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- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
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- Demonstration of InfluxDB and Grafana using a practice web application
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Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
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Lesson 9: The Product and Quotient Rules (slides)
1. Sec on 2.4
The Product and Quo ent Rules
V63.0121.011: Calculus I
Professor Ma hew Leingang
New York University
February 23, 2011
.
2. Announcements
Quiz 2 next week on
§§1.5, 1.6, 2.1, 2.2
Midterm March 7 on all
sec ons in class (covers
all sec ons up to 2.5)
3. Help!
Free resources:
Math Tutoring Center
(CIWW 524)
College Learning Center
(schedule on Blackboard)
TAs’ office hours
my office hours
each other!
4. Objectives
Understand and be able
to use the Product Rule
for the deriva ve of the
product of two func ons.
Understand and be able
to use the Quo ent Rule
for the deriva ve of the
quo ent of two
func ons.
5. Outline
Deriva ve of a Product
Deriva on
Examples
The Quo ent Rule
Deriva on
Examples
More deriva ves of trigonometric func ons
Deriva ve of Tangent and Cotangent
Deriva ve of Secant and Cosecant
More on the Power Rule
Power Rule for Nega ve Integers
6. Recollection and extension
We have shown that if u and v are func ons, that
(u + v)′ = u′ + v′
(u − v)′ = u′ − v′
What about uv?
7. Is the derivative of a product the
product of the derivatives?
(uv)′ = u′ v′ ?
.
8. Is the derivative of a product the
product of the derivatives?
(uv)′ = u′ v′ !
.
Try this with u = x and v = x2 .
9. Is the derivative of a product the
product of the derivatives?
(uv)′ = u′ v′ !
.
Try this with u = x and v = x2 .
Then uv = x3 =⇒ (uv)′ = 3x2 .
10. Is the derivative of a product the
product of the derivatives?
(uv)′ = u′ v′ !
.
Try this with u = x and v = x2 .
Then uv = x3 =⇒ (uv)′ = 3x2 .
But u′ v′ = 1 · 2x = 2x.
11. Is the derivative of a product the
product of the derivatives?
(uv)′ = u′ v′ !
.
Try this with u = x and v = x2 .
Then uv = x3 =⇒ (uv)′ = 3x2 .
But u′ v′ = 1 · 2x = 2x.
So we have to be more careful.
12. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
.
13. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
.
14. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.
.
15. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.
Say you get a 25 cent raise in
your hourly wages and work 5
hours more per week. How
much extra money do you
make?
.
16. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.
Say you get a 25 cent raise in
your hourly wages and work 5
hours more per week. How
much extra money do you
make?
.
∆I = 5 × $0.25 = $1.25?
17. Mmm...burgers
Say you work in a fast-food joint. You want to make more money.
What are your choices?
Work longer hours.
Get a raise.
Say you get a 25 cent raise in
your hourly wages and work 5
hours more per week. How
much extra money do you
make?
.
∆I = 5 × $0.25 = $1.25?
18. Money money money money
The answer depends on how much you work already and your
current wage. Suppose you work h hours and are paid w. You get a
me increase of ∆h and a wage increase of ∆w. Income is wages
mes hours, so
∆I = (w + ∆w)(h + ∆h) − wh
FOIL
= w · h + w · ∆h + ∆w · h + ∆w · ∆h − wh
= w · ∆h + ∆w · h + ∆w · ∆h
20. A geometric argument
Draw a box:
∆h w ∆h ∆w ∆h
h wh ∆w h
.
w ∆w
∆I = w ∆h + h ∆w + ∆w ∆h
21. Cash flow
Supose wages and hours are changing con nuously over me. Over
a me interval ∆t, what is the average rate of change of income?
∆I w ∆h + h ∆w + ∆w ∆h
=
∆t ∆t
∆h ∆w ∆h
=w +h + ∆w
∆t ∆t ∆t
22. Cash flow
Supose wages and hours are changing con nuously over me. Over
a me interval ∆t, what is the average rate of change of income?
∆I w ∆h + h ∆w + ∆w ∆h
=
∆t ∆t
∆h ∆w ∆h
=w +h + ∆w
∆t ∆t ∆t
What is the instantaneous rate of change of income?
dI ∆I dh dw
= lim =w +h +0
dt ∆t→0 ∆t dt dt
23. Eurekamen!
We have discovered
Theorem (The Product Rule)
Let u and v be differen able at x. Then
(uv)′ (x) = u(x)v′ (x) + u′ (x)v(x)
in Leibniz nota on
d du dv
(uv) = ·v+u
dx dx dx
25. Sanity Check
Example
Apply the product rule to u = x and v = x2 .
Solu on
(uv)′ (x) = u(x)v′ (x) + u′ (x)v(x) = x · (2x) + 1 · x2 = 3x2
This is what we get the “normal” way.
26. Which is better?
Example
Find this deriva ve two ways: first by direct mul plica on and then
by the product rule:
d [ ]
(3 − x2 )(x3 − x + 1)
dx
28. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by direct mul plica on:
d [ ] FOIL d [ 5 ]
(3 − x2 )(x3 − x + 1) = −x + 4x3 − x2 − 3x + 3
dx dx
29. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by direct mul plica on:
d [ ] FOIL d [ 5 ]
(3 − x2 )(x3 − x + 1) = −x + 4x3 − x2 − 3x + 3
dx dx
= −5x4 + 12x2 − 2x − 3
30. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
31. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
32. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
33. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
34. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
35. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
36. Which is better?
Example
d [ ]
(3 − x2 )(x3 − x + 1)
dx
Solu on
by the product rule:
( ) ( )
dy d d 3
= (3 − x ) (x − x + 1) + (3 − x )
2 3 2
(x − x + 1)
dx dx dx
= (−2x)(x3 − x + 1) + (3 − x2 )(3x2 − 1)
= −5x4 + 12x2 − 2x − 3
38. One more
Example
d
Find x sin x.
dx
Solu on
( ) ( )
d d d
x sin x = x sin x + x sin x
dx dx dx
39. One more
Example
d
Find x sin x.
dx
Solu on
( ) ( )
d d d
x sin x = x sin x + x sin x
dx dx dx
= 1 · sin x + x · cos x
40. One more
Example
d
Find x sin x.
dx
Solu on
( ) ( )
d d d
x sin x = x sin x + x sin x
dx dx dx
= 1 · sin x + x · cos x
= sin x + x cos x
41. Mnemonic
Let u = “hi” and v = “ho”. Then
(uv)′ = vu′ + uv′ = “ho dee hi plus hi dee ho”
42. Musical interlude
jazz bandleader and
singer
hit song “Minnie the
Moocher” featuring “hi
de ho” chorus
played Cur s in The Blues
Brothers
Cab Calloway
1907–1994
43. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.
44. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.
Solu on
(uvw)′ .
45. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.
Solu on
(uvw)′ = ((uv)w)′ .
46. Iterating the Product Rule
Example
Apply the product
Use the product rule to find the deriva ve of a three-fold product uvw.
rule to uv and w
Solu on
(uvw)′ = ((uv)w)′ .
47. Iterating the Product Rule
Example
Apply the product
Use the product rule to find the deriva ve of a three-fold product uvw.
rule to uv and w
Solu on
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
48. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.product
Apply the
rule to u and v
Solu on
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
49. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.product
Apply the
rule to u and v
Solu on
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
= (u′ v + uv′ )w + (uv)w′
50. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.
Solu on
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
= (u′ v + uv′ )w + (uv)w′
= u′ vw + uv′ w + uvw′
51. Iterating the Product Rule
Example
Use the product rule to find the deriva ve of a three-fold product uvw.
Solu on
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
= (u′ v + uv′ )w + (uv)w′
= u′ vw + uv′ w + uvw′
So we write down the product three mes, taking the deriva ve of each factor
once.
52. Outline
Deriva ve of a Product
Deriva on
Examples
The Quo ent Rule
Deriva on
Examples
More deriva ves of trigonometric func ons
Deriva ve of Tangent and Cotangent
Deriva ve of Secant and Cosecant
More on the Power Rule
Power Rule for Nega ve Integers
54. The Quotient Rule
What about the deriva ve of a quo ent?
u
Let u and v be differen able func ons and let Q = . Then
v
u = Qv
55. The Quotient Rule
What about the deriva ve of a quo ent?
u
Let u and v be differen able func ons and let Q = . Then
v
u = Qv
If Q is differen able, we have
u′ = (Qv)′ = Q′ v + Qv′
56. The Quotient Rule
What about the deriva ve of a quo ent?
u
Let u and v be differen able func ons and let Q = . Then
v
u = Qv
If Q is differen able, we have
u′ = (Qv)′ = Q′ v + Qv′
′ u′ − Qv′ u′ u v′
=⇒ Q = = − ·
v v v v
57. The Quotient Rule
What about the deriva ve of a quo ent?
u
Let u and v be differen able func ons and let Q = . Then
v
u = Qv
If Q is differen able, we have
u′ = (Qv)′ = Q′ v + Qv′
′ u′ − Qv′ u′ u v′
=⇒ Q = = − ·
v v v v
( u )′ u′ v − uv′
=⇒ Q′ = =
v v2
58. The Quotient Rule
What about the deriva ve of a quo ent?
u
Let u and v be differen able func ons and let Q = . Then
v
u = Qv
If Q is differen able, we have
u′ = (Qv)′ = Q′ v + Qv′
′ u′ − Qv′ u′ u v′
=⇒ Q = = − ·
v v v v
( u )′ u′ v − uv′
=⇒ Q′ = =
v v2
This is called the Quo ent Rule.
59. The Quotient Rule
We have discovered
Theorem (The Quo ent Rule)
u
Let u and v be differen able at x, and v(x) ̸= 0. Then is
v
differen able at x, and
( u )′ u′ (x)v(x) − u(x)v′ (x)
(x) =
v v(x)2
60. Verifying Example
Example
( )
d x2 d
Verify the quo ent rule by compu ng and comparing it to (x).
dx x dx
61. Verifying Example
Example
( )
d x2 d
Verify the quo ent rule by compu ng and comparing it to (x).
dx x dx
Solu on
( 2) ( )
d x x dx x2 − x2 dx (x) x · 2x − x2 · 1
d d
= =
dx x x2 x2
x2 d
= 2 =1= (x)
x dx
62. Mnemonic
Let u = “hi” and v = “lo”. Then
( u )′ vu′ − uv′
= = “lo dee hi minus hi dee lo over lo lo”
v v2
63. Examples
Example
d 2x + 5
1.
dx 3x − 2
d sin x
2.
dx x2
d 1
3. 2+t+2
dt t
82. Solution to second example
Solu on
d sin x x2 dx sin x − sin x
d
=
dx x2
83. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2
84. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
85. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
=
86. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2
=
87. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2 cos x
=
88. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2 cos x − 2x
=
89. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2 cos x − 2x sin x
=
90. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2 cos x − 2x sin x
=
x4
91. Solution to second example
Solu on
d sin x x2 dx sin x − sin x dx x2
d d
=
dx x2 (x2 )2
x2 cos x − 2x sin x
=
x4
x cos x − 2 sin x
=
x3
92. Another way to do it
Find the deriva ve with the product rule instead.
Solu on
d sin x d ( )
2
= sin x · x−2
dx x dx
( ) ( )
d −2 d −2
= sin x · x + sin x · x
dx dx
= cos x · x−2 + sin x · (−2x−3 )
= x−3 (x cos x − 2 sin x)
No ce the technique of factoring out the largest nega ve power,
leaving posi ve powers.
93. Examples
Example Answers
d 2x + 5 19
1. 1. −
dx 3x − 2 (3x − 2)2
d sin x x cos x − 2 sin x
2. 2.
dx x2 x3
d 1
3. 2+t+2
dt t
95. Solution to third example
Solu on
d 1 (t2 + t + 2)(0) − (1)(2t + 1)
=
dt t2 + t + 2 (t2 + t + 2)2
96. Solution to third example
Solu on
d 1 (t2 + t + 2)(0) − (1)(2t + 1)
=
dt t2 + t + 2 (t2 + t + 2)2
2t + 1
=− 2
(t + t + 2)2
97. A nice little takeaway
Fact
1
Let v be differen able at x, and v(x) ̸= 0. Then is differen able at
v
0, and ( )′
1 v′
=− 2
v v
Proof.
( )
d 1 v· d
dx (1)−1· d
dx v v · 0 − 1 · v′ v′
= = =− 2
dx v v2 v2 v
98. Examples
Example Answers
d 2x + 5 19
1. 1. −
dx 3x − 2 (3x − 2)2
d sin x x cos x − 2 sin x
2. 2.
dx x2 x3
d 1 2t + 1
3. 3. − 2
dt t2+t+2 (t + t + 2)2
99. Outline
Deriva ve of a Product
Deriva on
Examples
The Quo ent Rule
Deriva on
Examples
More deriva ves of trigonometric func ons
Deriva ve of Tangent and Cotangent
Deriva ve of Secant and Cosecant
More on the Power Rule
Power Rule for Nega ve Integers
101. Derivative of Tangent
Example
d
Find tan x
dx
Solu on
( )
d d sin x
tan x =
dx dx cos x
102. Derivative of Tangent
Example
d
Find tan x
dx
Solu on
( )
d d sin x cos x · cos x − sin x · (− sin x)
tan x = =
dx dx cos x cos2 x
103. Derivative of Tangent
Example
d
Find tan x
dx
Solu on
( )
d d sin x cos x · cos x − sin x · (− sin x)
tan x = =
dx dx cos x cos2 x
cos2 x + sin2 x
=
cos2 x
104. Derivative of Tangent
Example
d
Find tan x
dx
Solu on
( )
d d sin x cos x · cos x − sin x · (− sin x)
tan x = =
dx dx cos x cos2 x
cos2 x + sin2 x 1
= =
cos2 x cos2 x
105. Derivative of Tangent
Example
d
Find tan x
dx
Solu on
( )
d d sin x cos x · cos x − sin x · (− sin x)
tan x = =
dx dx cos x cos2 x
cos2 x + sin2 x 1
= 2x
= 2x
= sec2 x
cos cos
108. Derivative of Cotangent
Example
d
Find cot x
dx
Solu on
d d ( cos x ) sin x · (− sin x) − cos x · cos x
cot x = =
dx dx sin x sin2 x
109. Derivative of Cotangent
Example
d
Find cot x
dx
Solu on
d d ( cos x ) sin x · (− sin x) − cos x · cos x
cot x = =
dx dx sin x sin2 x
− sin2 x − cos2 x
=
sin2 x
110. Derivative of Cotangent
Example
d
Find cot x
dx
Solu on
d d ( cos x ) sin x · (− sin x) − cos x · cos x
cot x = =
dx dx sin x sin2 x
− sin2 x − cos2 x 1
= =− 2
sin2 x sin x
111. Derivative of Cotangent
Example
d
Find cot x
dx
Solu on
d d ( cos x ) sin x · (− sin x) − cos x · cos x
cot x = =
dx dx sin x sin2 x
− sin2 x − cos2 x 1
= = − 2 = − csc2 x
sin2 x sin x
113. Derivative of Secant
Example
d
Find sec x
dx
Solu on
( )
d d 1
sec x =
dx dx cos x
114. Derivative of Secant
Example
d
Find sec x
dx
Solu on
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
115. Derivative of Secant
Example
d
Find sec x
dx
Solu on
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x
=
cos2 x
116. Derivative of Secant
Example
d
Find sec x
dx
Solu on
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x 1 sin x
= = ·
cos2 x cos x cos x
117. Derivative of Secant
Example
d
Find sec x
dx
Solu on
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x 1 sin x
= = · = sec x tan x
cos2 x cos x cos x
120. Derivative of Cosecant
Example
d
Find csc x
dx
Solu on
( )
d d 1 sin x · 0 − 1 · (cos x)
csc x = =
dx dx sin x sin2 x
121. Derivative of Cosecant
Example
d
Find csc x
dx
Solu on
( )
d d 1 sin x · 0 − 1 · (cos x)
csc x = =
dx dx sin x sin2 x
cos x
=− 2
sin x
122. Derivative of Cosecant
Example
d
Find csc x
dx
Solu on
( )
d d 1 sin x · 0 − 1 · (cos x)
csc x = =
dx dx sin x sin2 x
cos x 1 cos x
=− 2 =− ·
sin x sin x sin x
123. Derivative of Cosecant
Example
d
Find csc x
dx
Solu on
( )
d d 1 sin x · 0 − 1 · (cos x)
csc x = =
dx dx sin x sin2 x
cos x 1 cos x
=− 2 =− · = − csc x cot x
sin x sin x sin x
124. Recap: Derivatives of
trigonometric functions
y y′
sin x cos x Func ons come in pairs
(sin/cos, tan/cot, sec/csc)
cos x − sin x
Deriva ves of pairs follow
tan x sec2 x similar pa erns, with
cot x − csc2 x func ons and
co-func ons switched
sec x sec x tan x and an extra sign.
csc x − csc x cot x
125. Outline
Deriva ve of a Product
Deriva on
Examples
The Quo ent Rule
Deriva on
Examples
More deriva ves of trigonometric func ons
Deriva ve of Tangent and Cotangent
Deriva ve of Secant and Cosecant
More on the Power Rule
Power Rule for Nega ve Integers
126. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
127. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
128. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
d −n d 1
x =
dx dx xn
129. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
d n
d −n d 1 dx x
x = =− n 2
dx dx xn (x )
130. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
d n
d −n d 1 dx x nxn−1
x = = − n 2 = − 2n
dx dx xn (x ) x
131. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
d n
d −n d 1 dx x nxn−1
x = n
= − n 2 = − 2n = −nxn−1−2n
dx dx x (x ) x
132. Power Rule for Negative Integers
We will use the quo ent rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for posi ve integers n.
Proof.
d n
d −n d 1 dx x nxn−1
x = n
= − n 2 = − 2n = −nxn−1−2n = −nx−n−1
dx dx x (x ) x
133. Summary
The Product Rule: (uv)′ = u′ v + uv′
( u )′ vu′ − uv′
The Quo ent Rule: =
v v2
Deriva ves of tangent/cotangent, secant/cosecant
d d
tan x = sec2 x sec x = sec x tan x
dx dx
d d
cot x = − csc2 x csc x = − csc x cot x
dx dx
The Power Rule is true for all whole number powers, including
nega ve powers:
d n
x = nxn−1
dx