We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
Lesson 16: Derivatives of Logarithmic and Exponential FunctionsMatthew Leingang
We show the the derivative of the exponential function is itself! And the derivative of the natural logarithm function is the reciprocal function. We also show how logarithms can make complicated differentiation problems easier.
Lesson 8: Derivatives of Polynomials and Exponential functionsMatthew Leingang
Some of the most famous rules of the calculus of derivatives: the power rule, the sum rule, the constant multiple rule, and the number e defined so that e^x is its own derivative!
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Matthew Leingang
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
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.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
The inverse of a function "undoes" the effect of the function. We look at the implications of that property in the derivative, as well as logarithmic functions, which are inverses of exponential functions.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
After defining the limit and calculating a few, we introduced the limit laws. Today we do the same for the derivative. We calculate a few and introduce laws which allow us to computer more. The Power Rule shows us how to compute derivatives of polynomials, and we can also find directly the derivative of sine and cosine.
The basic rules of differentiation, including the power rule, the sum rule, the constant multiple rule, and the rule for differentiating sine and cosine.
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.
Lesson 18: Indeterminate Forms and L'Hôpital's RuleMatthew Leingang
L'Hôpital's Rule is not a magic bullet (or a sledgehammer) but it does allow us to find limits of indeterminate forms such as 0/0 and ∞/∞. With some algebra we can use it to resolve other indeterminate forms such as ∞-∞ and 0^0.
Infinity is a tricky thing. It's tempting to treat it as a special number, but that can lead to trouble. In this slideshow we look at the different kinds of infinite limits and limits at infinity.
The chain rule helps us find a derivative of a composition of functions. It turns out that it's the product of the derivatives of the composed functions.
Continuous function have an important property that small changes in input do not produce large changes in output. The Intermediate Value Theorem shows that a continuous function takes all values between any two values. From this we know that your height and weight were once the same, and right now there are two points on opposite sides of the world with the same temperature!
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
the derivative measure the instantaneous rate of change of a function, or the slope of the line tangent to its graph. It has countless applications.
[Note: We did not do this entire show in class. We will finish it on Thursday]
We go over the trigonometric function, their inverses, and the derivatives of the inverse functions. The surprising fact is that these derivatives are simpler functions than the functions themselves.
When a quantity changes in proportion to itself (for instance, bacteria reproduction or radioactive decay), the growth or decay is exponential in nature. There are many many examples of this to be found.
Optimization is a killer feature of the derivative. Not only do we often want to optimize some system, nature does as well. We give a procedure and many examples.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Lesson 14: Derivatives of Logarithmic and Exponential Functions (slides)Mel Anthony Pepito
The exponential function is pretty much the only function whose derivative is itself. The derivative of the natural logarithm function is also beautiful as it fills in an important gap. Finally, the technique of logarithmic differentiation allows us to find derivatives without the product rule.
The second Fundamental Theorem of Calculus makes calculating definite integrals a problem of antidifferentiation!
(the slideshow has extra examples based on what happened in class)
Lesson 27: Integration by Substitution (Section 4 version)Matthew Leingang
The method of substitution is the chain rule in reverse. At first it looks magical, then logical, and then you realize there's an art to choosing the right substitution. We try to demystify with many worked-out examples and graphics.
A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
A function is continuous at a point if the limit of the function at the point equals the value of the function at that point. Another way to say it, f is continuous at a if values of f(x) are close to f(a) if x is close to a. This property has deep implications, such as this: right now there are two points on opposites sides of the world with the same temperature!
Streamlining assessment, feedback, and archival with auto-multiple-choiceMatthew Leingang
Auto-multiple-choice (AMC) is an open-source optical mark recognition software package built with Perl, LaTeX, XML, and sqlite. I use it for all my in-class quizzes and exams. Unique papers are created for each student, fixed-response items are scored automatically, and free-response problems, after manual scoring, have marks recorded in the same process. In the first part of the talk I will discuss AMC’s many features and why I feel it’s ideal for a mathematics course. My contributions to the AMC workflow include some scripts designed to automate the process of returning scored papers
back to students electronically. AMC provides an email gateway, but I have written programs to return graded papers via the DAV protocol to student’s dropboxes on our (Sakai) learning management systems. I will also show how graded papers can be archived, with appropriate metadata tags, into an Evernote notebook.
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
Lesson 24: Areas and Distances, The Definite Integral (handout)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.
Lesson 24: Areas and Distances, The Definite Integral (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.
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.
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.
There are various reasons why we would want to find the extreme (maximum and minimum values) of a function. Fermat's Theorem tells us we can find local extreme points by looking at critical points. This process is known as the Closed Interval Method.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
Honest Reviews of Tim Han LMA Course Program.pptxtimhan337
Personal development courses are widely available today, with each one promising life-changing outcomes. Tim Han’s Life Mastery Achievers (LMA) Course has drawn a lot of interest. In addition to offering my frank assessment of Success Insider’s LMA Course, this piece examines the course’s effects via a variety of Tim Han LMA course reviews and Success Insider comments.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
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Lesson 16: Derivatives of Logarithmic and Exponential Functions
1. Section 3.3
Derivatives of Exponential and
Logarithmic Functions
V63.0121.027, Calculus I
October 22, 2009
.
.
Image credit: heipei
. . . . . .
2. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
4. Derivatives of Exponential Functions
Fact
If f(x) = ax , then f′ (x) = f′ (0)ax .
Proof.
Follow your nose:
f(x + h) − f(x) a x+ h − a x
f′ (x) = lim = lim
h→0 h h→0 h
a x a h − ax a h−1
= lim = ax · lim = ax · f′ (0).
h→0 h h→0 h
. . . . . .
5. Derivatives of Exponential Functions
Fact
If f(x) = ax , then f′ (x) = f′ (0)ax .
Proof.
Follow your nose:
f(x + h) − f(x) a x+ h − a x
f′ (x) = lim = lim
h→0 h h→0 h
a x a h − ax a h−1
= lim = ax · lim = ax · f′ (0).
h→0 h h→0 h
To reiterate: the derivative of an exponential function is a
constant times that function. Much different from polynomials!
. . . . . .
6. The funny limit in the case of e
Remember the definition of e:
( )
1 n
e = lim 1 + = lim (1 + h)1/h
n→∞ n h→0
Question
eh − 1
What is lim ?
h→0 h
. . . . . .
7. The funny limit in the case of e
Remember the definition of e:
( )
1 n
e = lim 1 + = lim (1 + h)1/h
n→∞ n h→0
Question
eh − 1
What is lim ?
h→0 h
Answer
If h is small enough, e ≈ (1 + h)1/h . So
[ ]h
eh − 1 (1 + h)1/h − 1 (1 + h) − 1 h
≈ = = =1
h h h h
. . . . . .
8. The funny limit in the case of e
Remember the definition of e:
( )
1 n
e = lim 1 + = lim (1 + h)1/h
n→∞ n h→0
Question
eh − 1
What is lim ?
h→0 h
Answer
If h is small enough, e ≈ (1 + h)1/h . So
[ ]h
eh − 1 (1 + h)1/h − 1 (1 + h) − 1 h
≈ = = =1
h h h h
eh − 1
So in the limit we get equality: lim =1
h→0 h
. . . . . .
10. Exponential Growth
Commonly misused term to say something grows
exponentially
It means the rate of change (derivative) is proportional to the
current value
Examples: Natural population growth, compounded interest,
social networks
. . . . . .
11. Examples
Examples
Find these derivatives:
e3x
2
ex
x 2 ex
. . . . . .
12. Examples
Examples
Find these derivatives:
e3x
2
ex
x 2 ex
Solution
d 3x
e = 3e3x
dx
. . . . . .
13. Examples
Examples
Find these derivatives:
e3x
2
ex
x 2 ex
Solution
d 3x
e = 3e3x
dx
d x2 2 d 2
e = ex (x2 ) = 2xex
dx dx
. . . . . .
14. Examples
Examples
Find these derivatives:
e3x
2
ex
x 2 ex
Solution
d 3x
e = 3e3x
dx
d x2 2 d 2
e = ex (x2 ) = 2xex
dx dx
d 2 x
x e = 2xex + x2 ex
dx
. . . . . .
15. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
22. The Tower of Powers
y y′
The derivative of a
x3 3x2 power function is a
power function of one
x2 2x1
lower power
x1 1x0
x0 0
? ?
x−1 −1x−2
x−2 −2x−3
. . . . . .
23. The Tower of Powers
y y′
The derivative of a
x3 3x2 power function is a
power function of one
x2 2x1
lower power
x1 1x0 Each power function is
x 0
0 the derivative of another
power function, except
? x −1 x−1
x−1 −1x−2
x−2 −2x−3
. . . . . .
24. The Tower of Powers
y y′
The derivative of a
x3 3x2 power function is a
power function of one
x2 2x1
lower power
x1 1x0 Each power function is
x 0
0 the derivative of another
power function, except
ln x x −1 x−1
x−1 −1x−2 ln x fills in this gap
precisely.
x−2 −2x−3
. . . . . .
25. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
26. Other logarithms
Example
d x
Use implicit differentiation to find a.
dx
. . . . . .
27. Other logarithms
Example
d x
Use implicit differentiation to find a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
. . . . . .
28. Other logarithms
Example
d x
Use implicit differentiation to find a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
Differentiate implicitly:
1 dy dy
= ln a =⇒ = (ln a)y = (ln a)ax
y dx dx
. . . . . .
29. Other logarithms
Example
d x
Use implicit differentiation to find a.
dx
Solution
Let y = ax , so
ln y = ln ax = x ln a
Differentiate implicitly:
1 dy dy
= ln a =⇒ = (ln a)y = (ln a)ax
y dx dx
Before we showed y′ = y′ (0)y, so now we know that
2h − 1 3h − 1
ln 2 = lim ≈ 0.693 ln 3 = lim ≈ 1.10
h→0 h h→0 h
. . . . . .
31. Other logarithms
Example
d
Find loga x.
dx
Solution
Let y = loga x, so ay = x.
. . . . . .
32. Other logarithms
Example
d
Find loga x.
dx
Solution
Let y = loga x, so ay = x. Now differentiate implicitly:
dy dy 1 1
(ln a)ay = 1 =⇒ = y =
dx dx a ln a x ln a
. . . . . .
33. Other logarithms
Example
d
Find loga x.
dx
Solution
Let y = loga x, so ay = x. Now differentiate implicitly:
dy dy 1 1
(ln a)ay = 1 =⇒ = y =
dx dx a ln a x ln a
Another way to see this is to take the natural logarithm:
ln x
ay = x =⇒ y ln a = ln x =⇒ y =
ln a
dy 1 1
So = .
dx ln a x
. . . . . .
35. More examples
Example
d
Find log2 (x2 + 1)
dx
Answer
dy 1 1 2x
= 2+1
(2x) =
dx ln 2 x (ln 2)(x2 + 1)
. . . . . .
36. Outline
Derivative of the natural exponential function
Exponential Growth
Derivative of the natural logarithm function
Derivatives of other exponentials and logarithms
Other exponentials
Other logarithms
Logarithmic Differentiation
The power rule for irrational powers
. . . . . .
37. A nasty derivative
Example √
(x2 + 1) x + 3
Let y = . Find y′ .
x−1
. . . . . .
38. A nasty derivative
Example √
(x2 + 1) x + 3
Let y = . Find y′ .
x−1
Solution
We use the quotient rule, and the product rule in the numerator:
[ √ ] √
′ (x − 1) 2x x + 3 + (x2 + 1) 1 (x + 3)−1/2 − (x2 + 1) x + 3(1)
2
y =
(x − 1)2
√ √
2x x + 3 (x2 + 1) (x 2 + 1 ) x + 3
= + √ −
(x − 1 ) 2 x + 3(x − 1) (x − 1)2
. . . . . .
39. Another way
√
(x 2 + 1 ) x + 3
y=
x−1
1
ln y = ln(x2 + 1) + ln(x + 3) − ln(x − 1)
2
1 dy 2x 1 1
= 2 + −
y dx x + 1 2(x + 3) x − 1
So
( )
dy 2x 1 1
= + − y
dx x2 + 1 2(x + 3) x − 1
( ) √
2x 1 1 (x2 + 1) x + 3
= + −
x2 + 1 2(x + 3) x − 1 x−1
. . . . . .
40. Compare and contrast
Using the product, quotient, and power rules:
√ √
′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3
y = + √ −
(x − 1) 2 x + 3(x − 1) (x − 1)2
Using logarithmic differentiation:
( ) √
′ 2x 1 1 (x2 + 1) x + 3
y = + −
x2 + 1 2(x + 3) x − 1 x−1
. . . . . .
41. Compare and contrast
Using the product, quotient, and power rules:
√ √
′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3
y = + √ −
(x − 1) 2 x + 3(x − 1) (x − 1)2
Using logarithmic differentiation:
( ) √
′ 2x 1 1 (x2 + 1) x + 3
y = + −
x2 + 1 2(x + 3) x − 1 x−1
Are these the same?
. . . . . .
42. Compare and contrast
Using the product, quotient, and power rules:
√ √
′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3
y = + √ −
(x − 1) 2 x + 3(x − 1) (x − 1)2
Using logarithmic differentiation:
( ) √
′ 2x 1 1 (x2 + 1) x + 3
y = + −
x2 + 1 2(x + 3) x − 1 x−1
Are these the same?
Which do you like better?
. . . . . .
43. Compare and contrast
Using the product, quotient, and power rules:
√ √
′ 2x x + 3 (x2 + 1) (x2 + 1) x + 3
y = + √ −
(x − 1) 2 x + 3(x − 1) (x − 1)2
Using logarithmic differentiation:
( ) √
′ 2x 1 1 (x2 + 1) x + 3
y = + −
x2 + 1 2(x + 3) x − 1 x−1
Are these the same?
Which do you like better?
What kinds of expressions are well-suited for logarithmic
differentiation?
. . . . . .
44. Derivatives of powers
Let y = xx . Which of these is true?
(A) Since y is a power function, y′ = x · xx−1 = xx .
(B) Since y is an exponential function, y′ = (ln x) · xx
(C) Neither
. . . . . .
45. Derivatives of powers
Let y = xx . Which of these is true?
(A) Since y is a power function, y′ = x · xx−1 = xx .
(B) Since y is an exponential function, y′ = (ln x) · xx
(C) Neither
. . . . . .
46. It’s neither! Or both?
If y = xx , then
ln y = x ln x
1 dy 1
= x · + ln x = 1 + ln x
y dx x
dy
= xx + (ln x)xx
dx
Each of these terms is one of the wrong answers!
. . . . . .
48. Derivative of arbitrary powers
Fact (The power rule)
Let y = xr . Then y′ = rxr−1 .
Proof.
y = xr =⇒ ln y = r ln x
Now differentiate:
1 dy r
=
y dx x
dy y
=⇒ = r = rxr−1
dx x
. . . . . .