Lesson 9: The Product and Quotient Rules (slides)Matthew Leingang
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.
Lesson 27: Integration by Substitution (Section 4 version)Matthew Leingang
The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.
Lesson 27: Integration by Substitution (Section 10 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.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
Lesson 9: The Product and Quotient Rules (slides)Matthew Leingang
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.
Lesson 27: Integration by Substitution (Section 4 version)Matthew Leingang
The document outlines a calculus lecture on integration by substitution. It provides examples of using u-substitution to find antiderivatives of expressions like √(x^2+1) and tan(x). The key ideas are that if u is a function of x, its derivative du/dx can be used to rewrite the integrand and perform a u-substitution integration.
Lesson 27: Integration by Substitution (Section 10 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.
This document is the outline for a calculus class. It discusses the final exam date and review sessions, and outlines the topics of substitution for indefinite integrals and substitution for definite integrals. It provides an example of using substitution to find the integral of the square root of x^2 + 1 by letting u = x^2 + 1, and expresses this using both standard notation and Leibniz notation. It states the theorem of substitution rule.
Lesson 29: Integration by Substition (worksheet solutions)Matthew Leingang
This document contains the notes from a calculus class. It provides announcements about the final exam schedule and review sessions. It then discusses the technique of u-substitution for both indefinite and definite integrals. Examples are provided to illustrate how to use u-substitution to evaluate integrals involving trigonometric, polynomial, and other functions. The document emphasizes that u-substitution often makes evaluating integrals much easier than expanding them out directly.
The document discusses error propagation in physics measurements and calculations. It provides formulas for calculating the uncertainty (error) when adding, subtracting, multiplying and dividing measurements. The key points are:
1) When adding or subtracting measurements, the total uncertainty is the square root of the sum of the individual measurement uncertainties squared.
2) When multiplying or dividing measurements, the total uncertainty is calculated by adding the individual relative uncertainties squared.
3) For measurements raised to a power, the relative uncertainty is equal to the relative uncertainty of the measurement itself multiplied by the power.
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
The document provides examples demonstrating the chain rule for differentiating composite functions. The chain rule states that if y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx). Several examples are worked through applying the chain rule to functions composed of multiple operations like sin(x^2) or (x^2 + 5x - 1)^(2/3). The chain rule can be extended to chains of more than two functions as shown in later examples.
Lesson 27: Integration by Substitution, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
The document discusses velocity and acceleration in terms of position x. It provides equations showing that acceleration is equal to the derivative of velocity with respect to time, and the derivative of velocity with respect to x. It also gives examples of using these relationships to find velocity and position as functions of x and time for particles where acceleration is given.
This document provides examples and explanations for solving different types of word problems involving age, mixtures, clocks, variations, work, numbers, motion, and interest. It explains the general principles and formulas for each type of problem, and provides sample problems worked out step-by-step to illustrate the approaches. Key problem-solving techniques covered include using variables to represent unknown quantities, setting up and solving equations, and applying relevant formulas involving rates, times, percentages, and interest calculations.
The document provides examples and steps for solving linear equations and graphing lines from equations of the form y=mx+b. It includes 4 examples of solving linear equations for y and finding the slope and y-intercept of lines defined by equations. The examples are worked through step-by-step with the solutions shown.
The document describes the process of integration by partial fractions. It explains that when the degree of the numerator is greater than or equal to the denominator, division is performed. Otherwise, the denominator is factored. For each linear factor, the numerator is written as a sum of terms divided by that factor. For multiple linear factors, the numerator is written as a sum of terms divided by powers of that factor. Examples are provided to demonstrate these steps.
This document is from a Calculus I class at New York University and covers basic differentiation rules. It includes announcements about homework and a quiz. The objectives are to understand and use rules for differentiating constant functions, constants multiplied by functions, sums and differences of functions, and sine and cosine functions. Examples are provided of finding the derivatives of squaring, cubing, and square root functions using the definition of the derivative. Graphs and properties of derivatives are also discussed.
This document outlines and compares classical random walks, quantum random walks, and quantum random walks with memory. It introduces the key concepts of classical random walks on a line involving coin flips to move left or right. Quantum random walks are then described as taking place on a state space involving position and coin states, with transitions defined by unitary operators. Finally, quantum random walks with memory are discussed as generalizing the model to include memory of previous coin states.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
The product rule can be iterated to find the derivative of products with more than two factors. The derivative of a three-factor product uvw is u'vw + uv'w + uvw'. More generally, the derivative of a product of n factors breaks the product into a sum of n terms by applying the product rule recursively.
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.
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.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
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.
1. The document discusses algebraic principles for multiplying and factorizing sums and differences of numbers. It introduces the formula (a + b)(c + d) = ac + ad + bc + bd for multiplying two sums, and similar formulas for multiplying sums and differences.
2. It then applies these formulas to derive algebraic identities for the square of a sum, the difference of squares, and the product of a sum and difference. Examples are provided to demonstrate how these identities can be used to simplify calculations.
3. Readers are prompted with examples to practice applying the different algebraic formulas and identities introduced in the document.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
The document discusses implicit differentiation, which allows one to find the derivative of functions defined by a relation rather than explicitly. It gives an example of implicitly differentiating the relation x^2 + y^2 = 25 using the chain rule. Applying implicit differentiation and the chain rule derives the solution y = -x/y, which verifies the original implicit definition of the function.
The document discusses error propagation in physics measurements and calculations. It provides formulas for calculating the uncertainty (error) when adding, subtracting, multiplying and dividing measurements. The key points are:
1) When adding or subtracting measurements, the total uncertainty is the square root of the sum of the individual measurement uncertainties squared.
2) When multiplying or dividing measurements, the total uncertainty is calculated by adding the individual relative uncertainties squared.
3) For measurements raised to a power, the relative uncertainty is equal to the relative uncertainty of the measurement itself multiplied by the power.
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
The document provides examples demonstrating the chain rule for differentiating composite functions. The chain rule states that if y = g(u) and u = f(x), then dy/dx = (dy/du) * (du/dx). Several examples are worked through applying the chain rule to functions composed of multiple operations like sin(x^2) or (x^2 + 5x - 1)^(2/3). The chain rule can be extended to chains of more than two functions as shown in later examples.
Lesson 27: Integration by Substitution, part II (Section 10 version)Matthew Leingang
The document is the notes for a Calculus I class. It provides announcements for the upcoming class on Monday, which will involve reviewing course material rather than new topics. Examples are given of integration by substitution, including exponential, odd, and even functions. Multiple methods for substitutions are presented. The properties of odd and even functions are defined, and examples are shown graphically. Symmetric functions and their behavior under combinations are discussed.
The document discusses velocity and acceleration in terms of position x. It provides equations showing that acceleration is equal to the derivative of velocity with respect to time, and the derivative of velocity with respect to x. It also gives examples of using these relationships to find velocity and position as functions of x and time for particles where acceleration is given.
This document provides examples and explanations for solving different types of word problems involving age, mixtures, clocks, variations, work, numbers, motion, and interest. It explains the general principles and formulas for each type of problem, and provides sample problems worked out step-by-step to illustrate the approaches. Key problem-solving techniques covered include using variables to represent unknown quantities, setting up and solving equations, and applying relevant formulas involving rates, times, percentages, and interest calculations.
The document provides examples and steps for solving linear equations and graphing lines from equations of the form y=mx+b. It includes 4 examples of solving linear equations for y and finding the slope and y-intercept of lines defined by equations. The examples are worked through step-by-step with the solutions shown.
The document describes the process of integration by partial fractions. It explains that when the degree of the numerator is greater than or equal to the denominator, division is performed. Otherwise, the denominator is factored. For each linear factor, the numerator is written as a sum of terms divided by that factor. For multiple linear factors, the numerator is written as a sum of terms divided by powers of that factor. Examples are provided to demonstrate these steps.
This document is from a Calculus I class at New York University and covers basic differentiation rules. It includes announcements about homework and a quiz. The objectives are to understand and use rules for differentiating constant functions, constants multiplied by functions, sums and differences of functions, and sine and cosine functions. Examples are provided of finding the derivatives of squaring, cubing, and square root functions using the definition of the derivative. Graphs and properties of derivatives are also discussed.
This document outlines and compares classical random walks, quantum random walks, and quantum random walks with memory. It introduces the key concepts of classical random walks on a line involving coin flips to move left or right. Quantum random walks are then described as taking place on a state space involving position and coin states, with transitions defined by unitary operators. Finally, quantum random walks with memory are discussed as generalizing the model to include memory of previous coin states.
This document contains lecture notes on calculus including:
- Announcements about upcoming quizzes and exams
- An outline of topics to be covered including the derivative of a product, quotient rule, and power rule
- Examples of solving continuity problems using theorems
- A discussion of the derivative of a product of functions and how it is not simply the product of the individual derivatives
- Examples worked out step-by-step for understanding the concepts
The product rule can be iterated to find the derivative of products with more than two factors. The derivative of a three-factor product uvw is u'vw + uv'w + uvw'. More generally, the derivative of a product of n factors breaks the product into a sum of n terms by applying the product rule recursively.
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.
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.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
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.
1. The document discusses algebraic principles for multiplying and factorizing sums and differences of numbers. It introduces the formula (a + b)(c + d) = ac + ad + bc + bd for multiplying two sums, and similar formulas for multiplying sums and differences.
2. It then applies these formulas to derive algebraic identities for the square of a sum, the difference of squares, and the product of a sum and difference. Examples are provided to demonstrate how these identities can be used to simplify calculations.
3. Readers are prompted with examples to practice applying the different algebraic formulas and identities introduced in the document.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
In this section we look at problems where changing quantities are related. For instance, a growing oil slick is changing in diameter and volume at the same time. How are the rates of change of these quantities related? The chain rule for derivatives is the key.
The document discusses implicit differentiation, which allows one to find the derivative of functions defined by a relation rather than explicitly. It gives an example of implicitly differentiating the relation x^2 + y^2 = 25 using the chain rule. Applying implicit differentiation and the chain rule derives the solution y = -x/y, which verifies the original implicit definition of the function.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand the derivative of constants, the constant multiple rule, the sum rule, the difference rule, and derivatives of sine and cosine. Examples are provided to find the derivatives of squaring, cubing, square root, and cube root functions using the definition of the derivative. Graphs and properties of these functions and their derivatives are also discussed.
This document provides an outline for a calculus lecture on basic differentiation rules. It includes objectives to understand key rules like the constant multiple rule, sum rule, and derivatives of sine and cosine. Examples are worked through to find the derivatives of functions like squaring, cubing, square root, and cube root using the definition of the derivative. Graphs and properties of derived functions are also discussed.
This document provides an overview of linear systems and their application to economic models. It contains the following key points:
1) Wassily Leontief used systems of linear equations to model economies and make predictions, winning a Nobel Prize in Economics.
2) An example economic model is given with two industries (goods, services) and equations showing their internal demands.
3) The example forms equations to model the total demand and supply of each industry, and solves the system to find the dollar values each industry must produce.
4) Methods for solving systems of linear equations are discussed, including substitution and elimination approaches. The concepts of consistency and uniqueness of solutions are also introduced.
The document provides an overview of logic gates and Boolean algebra. It begins with a review of Boolean algebra concepts like variables being 1 or 0, NOT, OR, and AND operations. It then introduces basic logic gates like NOT, AND, OR, NAND, NOR, and XOR gates. It explains how to determine the output of logic circuits and write Boolean expressions as logic circuits. The document also covers converting between binary and decimal numbers, adding binary numbers using half adders and full adders, and briefly discusses flip-flops and memory.
The document is a summary of lecture notes for a Calculus I class. It discusses integration by substitution, providing theory, examples, and objectives. Key points covered include the substitution rule for indefinite integrals, working through examples like finding the integral of √x2+1 dx, and noting substitution can transform integrals into simpler forms. Definite integrals using substitution are also briefly mentioned.
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 document discusses using integration by parts to evaluate the integral x sin x dx. It chooses u(x) = x and v(x) = sin x, finds the derivatives u'(x) and v'(x), and uses the integration by parts formula to obtain x sin x dx = -x cos x + sin x. It also provides an example of using the trick of taking v(x) = 1 to evaluate ln x dx, choosing u(x) = ln x and v(x) = 1 and arriving at the solution ln x dx = x(ln x - 1) + C.
The document discusses the idea of integration by parts, which involves using the product rule in reverse to evaluate integrals that cannot be solved using other methods. It presents the integration by parts formula as u(x)v'(x)dx = u(x)v(x) - u'(x)v(x)dx and works through an example problem of evaluating the integral of xex dx using this formula. The example breaks the integral down into separate terms and shows that the overall integral equals xex - ex + C.
The document defines linear programming as a branch of mathematics used to find the optimal solution to problems with constraints. It provides examples of using linear programming to maximize profit or minimize costs in organizations. It also introduces drawing linear inequalities and solving simultaneous inequalities. The steps to formulate a linear programming problem are identified as defining variables and objectives, translating constraints, finding feasible solutions, and evaluating objectives to find optimal solutions.
Similar to Lesson 9: The Product and Quotient Rule (20)
This document provides guidance on developing effective lesson plans for calculus instructors. It recommends starting by defining specific learning objectives and assessments. Examples should be chosen carefully to illustrate concepts and engage students at a variety of levels. The lesson plan should include an introductory problem, definitions, theorems, examples, and group work. Timing for each section should be estimated. After teaching, the lesson can be improved by analyzing what was effective and what needs adjustment for the next time. Advanced preparation is key to looking prepared and ensuring students learn.
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.
This document discusses electronic grading of paper assessments using PDF forms. Key points include:
- Various tools for creating fillable PDF forms using LaTeX packages or desktop software.
- Methods for stamping completed forms onto scanned documents including using pdftk or overlaying in TikZ.
- Options for grading on tablets or desktops including GoodReader, PDFExpert, Adobe Acrobat.
- Extracting data from completed forms can be done in Adobe Acrobat or via command line with pdftk.
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
g(x) represents the area under the curve of f(t) between 0 and x.
.
x
What can you say about g? 2 4 6 8 10f
The First Fundamental Theorem of Calculus
Theorem (First Fundamental Theorem of Calculus)
Let f be a con nuous func on on [a, b]. Define the func on F on [a, b] by
∫ x
F(x) = f(t) dt
a
Then F is con nuous on [a, b] and differentiable on (a, b) and for all x in (a, b),
F′(x
Lesson 26: The Fundamental Theorem of Calculus (slides)Matthew Leingang
The document discusses the Fundamental Theorem of Calculus, which has two parts. The first part states that if a function f is continuous on an interval, then the derivative of the integral of f is equal to f. This is proven using Riemann sums. The second part relates the integral of a function f to the integral of its derivative F'. Examples are provided to illustrate how the area under a curve relates to these concepts.
Lesson 27: Integration by Substitution (handout)Matthew Leingang
This document contains lecture notes on integration by substitution from a Calculus I class. It introduces the technique of substitution for both indefinite and definite integrals. For indefinite integrals, the substitution rule is presented, along with examples of using substitutions to evaluate integrals involving polynomials, trigonometric, exponential, and other functions. For definite integrals, the substitution rule is extended and examples are worked through both with and without first finding the indefinite integral. The document emphasizes that substitution often simplifies integrals and makes them easier to evaluate.
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1) The document discusses lecture notes on Section 5.4: The Fundamental Theorem of Calculus from a Calculus I course. 2) It covers stating and explaining the Fundamental Theorems of Calculus and using the first fundamental theorem to find derivatives of functions defined by integrals. 3) The lecture outlines the first fundamental theorem, which relates differentiation and integration, and gives examples of applying it.
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2) An example of graphing the cubic function f(x) = 2x^3 - 3x^2 - 12x through analyzing its derivatives.
3) Explanations of the increasing/decreasing test and concavity test to determine monotonicity and concavity from a function's derivatives.
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Lesson 9: The Product and Quotient Rule
1. Section 2.4
The Product and Quotient Rules
V63.0121.027, Calculus I
October 1, 2009
Announcements
Quiz 2 is next week, covering §§1.4–2.1
Midterm I is October 15, covering §§1.1–2.4 (today)
Office Hours today 3–4, check website for current
. . . . . .
2. Outline
The Product Rule
Derivation
Examples
The Quotient Rule
Derivation
Examples
More derivatives of trigonometric functions
Derivative of Tangent
Derivative of Secant and Cosecant
More on the Power Rule
Power Rule for Positive Integers by Induction
Power Rule for Negative Integers
. . . . . .
13. 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?
. .
. . . . . .
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.
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?
∆
. . . . . .
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?
. I = 5 × $0..25 = $1.25?
∆
. . . . . .
16. 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 time increase of ∆h and a wage increase of ∆w. Income is
wages times hours, so
∆I = (w + ∆w)(h + ∆h) − wh
FOIL
= w · h + w · ∆h + ∆w · h + ∆w · ∆h − wh
= w · ∆h + ∆ w · h + ∆ w · ∆h
. . . . . .
17. A geometric argument
Draw a box:
. h
∆ w
. ∆h . w ∆h
∆
h
. w
. h . wh
∆
.
w
. . w
∆
. . . . . .
18. A geometric argument
Draw a box:
. h
∆ w
. ∆h . w ∆h
∆
h
. w
. h . wh
∆
.
w
. . w
∆
∆I = w ∆h + h ∆w + ∆ w ∆h
. . . . . .
21. Eurekamen!
We have discovered
Theorem (The Product Rule)
Let u and v be differentiable at x. Then
(uv)′ (x) = u(x)v′ (x) + u′ (x)v(x)
in Leibniz notation
d du dv
(uv) = ·v+u
dx dx dx
. . . . . .
23. Example
Apply the product rule to u = x and v = x2 .
Solution
(uv)′ (x) = u(x)v′ (x) + u′ (x)v(x) = x · (2x) + 1 · x2 = 3x2
This is what we get the “normal” way.
. . . . . .
35. One more
Example
d
Find x sin x.
dx
Solution
( ) ( )
d d d
x sin x = x sin x + x sin x
dx dx dx
. . . . . .
36. One more
Example
d
Find x sin x.
dx
Solution
( ) ( )
d d d
x sin x = x sin x + x sin x
dx dx dx
= 1 · sin x + x · cos x
. . . . . .
37. One more
Example
d
Find x sin x.
dx
Solution
( ) ( )
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
. . . . . .
38. Mnemonic
Let u = “hi” and v = “ho”. Then
(uv)′ = vu′ + uv′ = “ho dee hi plus hi dee ho”
. . . . . .
39. Musical interlude
jazz bandleader and
singer
hit song “Minnie the
Moocher” featuring “hi
de ho” chorus
played Curtis in The
Blues Brothers
Cab Calloway
1907–1994
. . . . . .
40. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw.
. . . . . .
41. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw.
Solution
(uvw)′ .
. . . . . .
42. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw.
Solution
(uvw)′ = ((uv)w)′ .
. . . . . .
43. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw. .
Apply the product rule
to uv and w
Solution
(uvw)′ = ((uv)w)′ .
. . . . . .
44. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw. .
Apply the product rule
to uv and w
Solution
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
. . . . . .
45. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw. .
Apply the product rule
to u and v
Solution
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
. . . . . .
46. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw. .
Apply the product rule
to u and v
Solution
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
= (u′ v + uv′ )w + (uv)w′
. . . . . .
47. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw.
Solution
(uvw)′ = ((uv)w)′ . = (uv)′ w + (uv)w′ .
= (u′ v + uv′ )w + (uv)w′
= u′ vw + uv′ w + uvw′
. . . . . .
48. Iterating the Product Rule
Example
Use the product rule to find the derivative of a three-fold product
uvw.
Solution
(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 times, taking the derivative
of each factor once.
. . . . . .
49. Outline
The Product Rule
Derivation
Examples
The Quotient Rule
Derivation
Examples
More derivatives of trigonometric functions
Derivative of Tangent
Derivative of Secant and Cosecant
More on the Power Rule
Power Rule for Positive Integers by Induction
Power Rule for Negative Integers
. . . . . .
53. The Quotient Rule
What about the derivative of a quotient?
u
Let u and v be differentiable functions and let Q = . Then
v
u = Qv
If Q is differentiable, we have
u′ = (Qv)′ = Q′ v + Qv′
u′ − Qv′ u′ u v′
=⇒ Q′ = = − ·
v v v v
. . . . . .
54. The Quotient Rule
What about the derivative of a quotient?
u
Let u and v be differentiable functions and let Q = . Then
v
u = Qv
If Q is differentiable, we have
u′ = (Qv)′ = Q′ v + Qv′
u′ − Qv′ u′ u v′
=⇒ Q′ = = − ·
v v v v
( u )′ u′ v − uv′
=⇒ Q′ = =
v v2
. . . . . .
55. The Quotient Rule
What about the derivative of a quotient?
u
Let u and v be differentiable functions and let Q = . Then
v
u = Qv
If Q is differentiable, 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 Quotient Rule.
. . . . . .
56. Verifying Example
Example ( )
d x2
Verify the quotient rule by computing and comparing it
dx x
d
to (x).
dx
. . . . . .
57. Verifying Example
Example ( )
d x2
Verify the quotient rule by computing and comparing it
dx x
d
to (x).
dx
Solution
( ) d
( ) d
d x2 x dx x2 − x2 dx (x)
=
dx x x2
x · 2x − x2 · 1
=
x2
x 2 d
= 2 =1= (x)
x dx
. . . . . .
58. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x2 − 1
d t−1
3.
dt t2 + t + 2
. . . . . .
79. Solution to third example
d t−1 (t2 + t + 2)(1) − (t − 1)(2t + 1)
=
dt t2 + t + 2 (t2 + t + 2)2
. . . . . .
80. Solution to third example
d t−1 (t2 + t + 2)(1) − (t − 1)(2t + 1)
=
dt t2 + t + 2 (t2 + t + 2)2
(t2 + t + 2) − (2t2 − t − 1)
=
(t2 + t + 2)2
−t2 + 2t + 3
= 2
(t + t + 2)2
. . . . . .
81. Examples
Example
d 2x + 5
1.
dx 3x − 2
d 2x + 1
2.
dx x2 − 1
d t−1
3.
dt t2 + t + 2
Answers
19
1. −
(3x − 2)2
( )
2 x2 + x + 1
2. −
(x2 − 1)2
−t2 + 2t + 3
3. 2
(t2 + t + 2)
. . . . . .
82. Mnemonic
Let u = “hi” and v = “lo”. Then
( u )′ vu′ − uv′
= = “lo dee hi minus hi dee lo over lo lo”
v v2
. . . . . .
83. Outline
The Product Rule
Derivation
Examples
The Quotient Rule
Derivation
Examples
More derivatives of trigonometric functions
Derivative of Tangent
Derivative of Secant and Cosecant
More on the Power Rule
Power Rule for Positive Integers by Induction
Power Rule for Negative Integers
. . . . . .
85. Derivative of Tangent
Example
d
Find tan x
dx
Solution
( )
d d sin x
tan x =
dx dx cos x
. . . . . .
86. Derivative of Tangent
Example
d
Find tan x
dx
Solution
( )
d d sin x cos x · cos x − sin x · (− sin x)
tan x = =
dx dx cos x cos2 x
. . . . . .
87. Derivative of Tangent
Example
d
Find tan x
dx
Solution
( )
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
. . . . . .
88. Derivative of Tangent
Example
d
Find tan x
dx
Solution
( )
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
=
cos cos2 x
. . . . . .
89. Derivative of Tangent
Example
d
Find tan x
dx
Solution
( )
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
= = sec2 x
cos cos2 x
. . . . . .
93. Derivative of Secant
Example
d
Find sec x
dx
Solution
( )
d d 1
sec x =
dx dx cos x
. . . . . .
94. Derivative of Secant
Example
d
Find sec x
dx
Solution
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
. . . . . .
95. Derivative of Secant
Example
d
Find sec x
dx
Solution
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x
=
cos2 x
. . . . . .
96. Derivative of Secant
Example
d
Find sec x
dx
Solution
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x 1 sin x
= 2x
= ·
cos cos x cos x
. . . . . .
97. Derivative of Secant
Example
d
Find sec x
dx
Solution
( )
d d 1 cos x · 0 − 1 · (− sin x)
sec x = =
dx dx cos x cos2 x
sin x 1 sin x
= 2x
= · = sec x tan x
cos cos x cos x
. . . . . .
99. Derivative of Cosecant
Example
d
Find csc x
dx
Answer
d
csc x = − csc x cot x
dx
. . . . . .
100. Recap: Derivatives of trigonometric functions
y y′
Functions come in pairs
sin x cos x
(sin/cos, tan/cot, sec/csc)
cos x − sin x Derivatives of pairs
tan x sec x 2 follow similar patterns,
with functions and
cot x − csc2 x co-functions switched
sec x sec x tan x and an extra sign.
csc x − csc x cot x
. . . . . .
101. Outline
The Product Rule
Derivation
Examples
The Quotient Rule
Derivation
Examples
More derivatives of trigonometric functions
Derivative of Tangent
Derivative of Secant and Cosecant
More on the Power Rule
Power Rule for Positive Integers by Induction
Power Rule for Negative Integers
. . . . . .
104. Principle of Mathematical Induction
.
Suppose S(1) is
true and S(n + 1)
is true whenever
.
S(n) is true. Then
S(n) is true for all
n.
.
.
Image credit: Kool Skatkat
. . . . . .
105. Power Rule for Positive Integers by Induction
Theorem
Let n be a positive integer. Then
d n
x = nxn−1
dx
Proof.
By induction on n. We can show it to be true for n = 1 directly.
. . . . . .
106. Power Rule for Positive Integers by Induction
Theorem
Let n be a positive integer. Then
d n
x = nxn−1
dx
Proof.
By induction on n. We can show it to be true for n = 1 directly.
d n
Suppose for some n that x = nxn−1 . Then
dx
d n +1 d
x = (x · xn )
dx dx
. . . . . .
107. Power Rule for Positive Integers by Induction
Theorem
Let n be a positive integer. Then
d n
x = nxn−1
dx
Proof.
By induction on n. We can show it to be true for n = 1 directly.
d n
Suppose for some n that x = nxn−1 . Then
dx
d n +1 d
x = (x · xn )
dx dx )
( ( )
d n d n
= x x +x x
dx dx
. . . . . .
108. Power Rule for Positive Integers by Induction
Theorem
Let n be a positive integer. Then
d n
x = nxn−1
dx
Proof.
By induction on n. We can show it to be true for n = 1 directly.
d n
Suppose for some n that x = nxn−1 . Then
dx
d n +1 d
x = (x · xn )
dx dx )
( ( )
d n d n
= x x +x x
dx dx
= 1 · xn + x · nxn−1 = (n + 1)xn
. . . . . .
109. Power Rule for Negative Integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for positive integers n.
. . . . . .
110. Power Rule for Negative Integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for positive integers n.
Proof.
d −n d 1
x =
dx dx xn
. . . . . .
111. Power Rule for Negative Integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for positive integers n.
Proof.
d −n d 1
x =
dx dx xn
d d n
xn · dx 1 − 1 · dx x
=
x2n
. . . . . .
112. Power Rule for Negative Integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for positive integers n.
Proof.
d −n d 1
x =
dx dx xn
d d n
xn · dx 1 − 1 · dx x
=
x2n
0 − nx n −1
=
x2n
. . . . . .
113. Power Rule for Negative Integers
Use the quotient rule to prove
Theorem
d −n
x = (−n)x−n−1
dx
for positive integers n.
Proof.
d −n d 1
x =
dx dx xn
d d
xn · dx 1 − 1 · dx xn
=
x2n
0 − nx n −1
= = −nx−n−1
x2n
. . . . . .
114. What have we learned today?
The Product Rule: (uv)′ = u′ v + uv′
( u )′ vu′ − uv′
The Quotient Rule: =
v v2
Derivatives 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 negative powers:
d n
x = nxn−1
dx
. . . . . .