Topics for Test #2 (Chapter 3 Sections 3.1-3.8, Exponentials)
CHAPTER 3 DERIVATIVES
Derivative rules for all the standard functions of calculus are established, including general
results and techniques, such as the Chain Rule and implicit differentiation. In addition, we begin
to apply the derivative to problems of a practical nature: interpreting the derivative as a rate of
change, and related rates problems.
3.1 Introducing the Derivative
Overview
The derivative of a function f is defined as a limit, and interpreted as both the slope of lines
tangent to the graph of f and the instantaneous rate of change of f. The connection between
differentiability and continuity is presented, and we also explain the conditions under which a
function is not differentiable at a point.
Lecture
This section is of fundamental importance for the entire chapter. The limit definition of the
derivative is used repeatedly in coming sections to develop rules of differentiation. Here is a
brief recap of the ideas presented in Section 2.1 to arrive at the definition of the average rate of
change of f over [a, x] and the slope of the tangent line at the point x = a :
𝑓(𝑥)−𝑓(𝑎)
𝑓(𝑥)−𝑓(𝑎)
𝑚𝑠𝑒𝑐 = 𝑥−𝑎 and 𝑚𝑡𝑎𝑛 = lim 𝑥−𝑎
𝑥→𝑎
Figure 3.3 in the text is the first of several figures that uses the same general layout to convey
ideas about the derivative. After studying the figure, do Example 1.
Now, watch the Video Presentation (Tangents and Derivatives) in the 3.1 Video Lecture link in
the Interactive e-book.
From figure 3.5 in the Interactive e-book, you obtain the alternative definitions
𝑓(𝑎+ℎ)−𝑓(𝑎)
𝑓(𝑎+ℎ)−𝑓(𝑎)
𝑚𝑠𝑒𝑐 =
and 𝑚𝑡𝑎𝑛 = lim
. Do Example 2.
ℎ
ℎ
ℎ→0
These definitions lead to a new function, the derivative. Replacing ‘a’ with the variable ‘x’
𝑓(𝑥+ℎ)−𝑓(𝑥)
produces the limit definition of the derivative𝑓 ′ (𝑥) = lim
. Do Example 3.
ℎ
ℎ→0
Now, watch the Video Presentation (Derivative as a Function) in the 2.6 Video Lecture link in
the Interactive e-book.
You should become familiar with the varied notation used to denote the derivative. Study
Examples 4 and 5.
Explore the relationship between the graph of a function and the graph of its derivative in
Examples 6 and 7.
Go over Theorem 3.1 and Theorem 3.1 (Alternative Version).
There are various ways the derivative can fail to exist at a point. See Example 8.
3.2 Rules of Differentiation
Overview
We begin an informative campaign of deriving formulas for the derivatives of the standard
functions of calculus. By section’s end, you will be able to find derivatives of polynomial
functions by combining the power, constant multiple, and sum rules. Additionally, higher order
derivatives are introduced.
Lecture
More than 25 derivative formulas, both general and function-specific, are included in this
chapter; most of them are essential for subsequent chapters.
The time to begin memorizing the formulas is now, not at the end of the chapter.
Now, watch the Video Presentation (Derivative Rules) in the 3.2 Video Lecture link in the
Interactive e-book.
Theorem 3.2 presents the Constant Rule; see Figure 3.20
Theorem 3.3 presents the Power Rule for nonnegative integers. Notice that the Constant Rule is a
special case of the Power Rule. Do Example 1.
Theorem 3.4 presents the Constant Multiple Rule. Do Example 2.
Theorem 3.5 presents the Sum Rule. Do Example 3.
Find slopes and equations of tangent lines in Example 4.
Practice finding higher order derivatives in Example 5.
3.3 The Product and Quotient Rules
Overview
A standard treatment of the Product and Quotient Rules is given, and the Quotient Rule is used to
extend the Power Rule to negative integers.
Lecture
If one had to identify the top ten student errors when performing differentiation, surely the
failure to use the Product and Quotient Rules correctly would be at or near the top of the list.
Be careful around products and quotients.
Theorem 3.6 presents the Product Rule. The theorem states that the derivative of the product of
two functions is the derivative of the first function multiplied by the second function plus the
first function multiplied by the derivative of the second function.
Caution: the derivative of the product of two functions is not the product of the derivatives.
Work Example 1, and then go to www.mathtv.com . In the Videos by Topic section, scroll down
to Calculus, and open Derivatives-Product Rule.
Theorem 3.7 presents the Quotient Rule. The theorem states that the derivative of the quotient of
two functions equal the denominator multiplied by the derivative of the numerator minus the
numerator multiplied by the derivative of the denominator, all divided by the denominator
squared.
Caution: the derivative of the quotient of two functions is not the quotient of the derivatives.
Work Example 2 and 3, and then go to www.mathtv.com . In the Videos by Topic section,
scroll down to Calculus, and open Derivatives-Quotient Rule.
Theorem 3.8 (Extended Power Rule), generalize the Power Rule to all integer powers (positive
or negative) See Example 4
Memorizing the Product and Quotient Rules is much easier with a mnemonic device. Popular
examples include D1⋅ 2 +1⋅D2 for the Product Rule and (LoDHi –HiDLo)/ Lo2 for the Quotient
Rule.
Example 6, combines derivative rules. Notice that if you multiply the factors, in this example, in
the numerator before taking the derivatives, you will obtain the same result by only using the
quotient rule.
Avoid using the Product or Quotient Rules when it is unnecessary.
As an example, if f (x) = x2 (x − 3), do not use the Product Rule to find the derivative. Instead
multiply the factors first, and then use the Power and Sum Rules to find the derivative.
4−𝑥 3
If 𝑓(𝑥) = 2 , do not use the Quotient Rule. Instead divide (or rewrite the function as 𝑓(𝑥) =
𝑥
(4 − 𝑥 3 )𝑥 −2), and find the derivative as you did in the previous example.
3.4 Derivatives of Trigonometric Functions
Overview
We continue our journey of discovering the derivatives of the standard families of functions, this
time turning our attention to trigonometric functions.
Lecture
Theorem 3.9 is needed in order to derive the formulas for the derivatives of sin x and cos x. Skip
the proof of the theorem. Apply Theorem 3.9 in Example 1. Find derivatives using trigonometric
functions in Example 2.
Find the derivative of the remaining trigonometric functions by employing the Quotient Rule in
Example 3. Find the derivative of other Trigonometric functions in Example 4, and find higher
order derivatives in Example 5.
Watch the first 13:28 minutes of the Video Presentation (Derivative of Trigonometric Functions)
in the 3.4 Video Lecture link in the Interactive e-book.
3.5 Derivatives as Rates of Change
Overview
We take a break from presenting derivative rules to focus on the extremely important
interpretation of the derivative as a rate of change.
Lecture
The interpretation of the derivative as a rate of change leads to the common characterization of
calculus as the mathematics of change.
The study of motion and calculus goes hand-in-hand, and this section has a thorough discussion
of rectilinear motion, explaining the relationships between position, velocity, speed, and
acceleration. See Examples 1through 3.
Now, watch the Video Presentation (Derivatives as Rates of Change) in the 3.5 Video Lecture
link in the Interactive e-book.
3.6 The Chain Rule
Overview
The Chain Rule is introduced, both in differential form using Leibniz notation and using
functional notation. The Chain Rule is a method of differentiating composite functions.
Lecture
The text offers an anecdote about the rate of picking apples that will help explain why the
derivative of a composition of functions is the product of derivatives see figure 3.42.
Study Theorem 3.12: The Chain Rule
Notice that when using Version 1 of the Chain Rule, it’s easy to visually confirm that the
𝑑𝑦
𝑑𝑦 𝑑𝑢
formula has been written correctly because the du “cancels” in 𝑑𝑥 = 𝑑𝑢 𝑑𝑥
When using Version 2, the derivative of a composition of functions f(g(x)) is the derivative of the
function f times the derivative of g.
As an example: the derivative sin (x2 + x) is the derivative of sin(x2 + x) times the derivative of
(x2 + x). So the derivative sin (x2 + x) = cos (x2 + x)*(2x+1).
Now, watch the Video Presentation (The Chain Rule) in the 3.6 Video Lecture link in the
Interactive e-book.
Review the Guidelines for Using the Chain Rule with Examples 1through 5.
3.7 Implicit Differentiation
Overview
In order to find the derivative of functions defined implicitly, we present the technique of
implicit differentiation. This technique is also used to prove an important extension of the Power
Rule: We show that the Power Rule for rational exponents is the same as the Power Rule for
integer exponents.
Lecture
Certain curves appear to have well-defined tangent lines throughout, and yet we do not know at
present how to compute dy / dx because it is difficult or impossible to solve for y. In those cases,
we must use implicit differentiation. Implicit differentiation works by assuming that an equation
in x and y does, in fact, define one or more functions y = f (x). The conditions for which this
assumption is true are given in the Implicit Function Theorem, covered in more advanced
courses. We replace all occurrences of y in a given equation with the function y(x), and
differentiate with respect to x. The Chain Rule is needed whenever the variable y is composed
with another function. Most calculations of dy / dx using the method of implicit differentiation
result in a formula involving both x and y, which means that both coordinates of a point are
needed to find the slope of a tangent line at that point.
Now, watch the Video Presentation (Implicit Differentiation) in the 3.7 Video Lecture link in the
Interactive e-book, and do Examples 1through 3.
Study Theorem 3.14, which extends to Power Rule to rational exponents, and do Example 4 and
5.
3.8 Related Rates
Overview
Given a relationship between two or more variables that change in time t, we can differentiate
the relationship with respect to t to produce a related rates equation. The basic idea in a typical
related rates problem is to use the related rates equation to solve for one rate of change in terms
of the other rates of change.
Lecture
In these types of problems it is essential to identify the rates that are known, and the rate that we
seek. Once past this step, an equation (sometimes more than one) that relates the variables in
question is needed. This equation is then differentiated to produce the related rates equation,
from which the solution can be found. In all but the simplest of related rates problems, the Chain
Rule is in play; in some cases implicit differentiation provides the most efficient means of
producing the related rates equation.
Now, watch the Video Presentation (Related Rates) in the 3.8 Video Lecture link in the
Interactive e-book, and do Examples 1through 4. The more examples you see and practice, the
better you get at it.
Chapter 3 (Sections 3.1-3.8) Key Terms and Concepts
Average and instantaneous rate of change (Section 3.1)
The derivative as instantaneous rate of change and slopes of tangent lines (Section 2.1)
Two derivative formulas (Section 3.1)
Differentiable implies continuous (Theorem 3.1) (Section 3.1)
Constant Rule (Theorem 3.2) (Section 3.2)
Power Rule (Theorem 3.3) (Section 3.2)
Constant Multiple Rule (Theorem 3.4) (Section 3.2)
Sum Rule (Theorem 3.5) (Section 3.2)
Generalized Sum and Difference Rule (Section 3.2)
Higher-order derivatives (Section 3.2)
Product Rule (Theorem 3.6) (Section 3.3)
Quotient Rule (Theorem 3.7) (Section 3.3)
Extended Power Rule (Theorem 3.8) (Section 3.3)
Important trigonometric limits (Theorem 3.9) (Section 3.4)
Derivatives of sin x and cos x (Theorem 3.10) (Section 3.4)
Derivatives of the trigonometric functions (Theorem 3.11) (Section 3.4)
Position and velocity (Section 3.5)
Speed and acceleration (Section 3.5)
The Chain Rule (Theorem 3.12) (Section 3.6)
Chain Rule for powers (Theorem 3.13) (Section 3.6)
Implicit differentiation (Section 3.7)
Power Rule for rational exponents (Theorem 3.14) (Section 3.7)
General Procedure for Related Rate Problems (Section 3.8)
Study the Exponential Handout link