Category Archives: Derivatives

Why Techniques for Differentiation?

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You have learned and used formulas for finding the derivatives of the Elementary Functions. These can be applied to functions made up of the Elementary Functions and extended to other expressions.

Many functions are made by combining the Elementary Functions. For example, polynomials are the sum and differences of a constants and powers of x multiplied by constants (their coefficients).

The functions you will look at next are the sums, differences, products, quotients, and/or composites of the Elementary Function. You will learn five techniques for handling these.

Sums and differences are found by differentiating the individual terms. Then there are the Product Rule for products of functions, the Quotient Rule for quotients and the Chain Rule for compositions. The techniques are often used in combinations.

Learn to see the patterns in the functions and learn what procedure to use for each.

HINT: Memorize the techniques as you learn them. After all these years, I still say the formulas and techniques as I use them. So, for products I repeat in my mind “the first times the derivative of the second plus the second times the derivative of the first” as I do the computation. Forget about mnemonics – just say the technique as you use it, and you’ll memorize it easily. 

Implicit relations and inverses have their own techniques; they use the basic formulas and techniques in different ways.

Since derivatives are functions, they have their own derivatives. The derivative of the first derivative is called the second derivative. Then there is the third derivative, the fourth derivative and so on. Mostly, you will use only the first three. No new rules to learn; higher order derivatives are computed the same way as the first derivative, and in fact, they often get simpler.

The proofs of the formulas and techniques (they are really theorems) are interesting from a mathematical point of view. You should follow along when your teacher shows them to you so that you understand why they work and where they come from. You will not be asked to reproduce the proofs on the AP Calculus Exam.

AP Calculus Course and Exam Description Unit 3 Sections 2.8 – 2.10 and Unit 3 all

Why Radian Measure?

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I’m sure you’ve been wondering since you first heard about radian measure before calculus why calculus is always done in radian measure. Once you’ve learned the derivatives of the trigonometric functions, you will appreciate why radians are pretty much the only choice for calculus people.

The idea for this series of posts, The Why Series, a this post I wrote a few years ago called “Why Radians?”  The post gets a lot of ‘hits’ every year. While that post was written for teachers, there is no reason you, a student, shouldn’t read it as well; it’s not a secret. Or maybe it is, but now that you’re initiated into calculus, you may read it.

If you’re still wondering why calculus people use radians, follow this link: Why Radians?

AP Calculus Course and Exam Description Unit 2 Section 7.

Why Formulas for Derivatives

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That’s pretty obvious: Finding derivatives using limits is a pain!

Knowing the derivatives of the common functions makes it easy to find.

No hiding it: you need to memorize these formulas. The best way is to learn them as you get them, a few at a time. I’m sure your teacher will not give them to you all at once. From the first day, memorize them by saying them to yourself as you use them. Don’t wait until the night before the test.

It’s really not too bad: there are only about seventeen formulas for the derivatives of the Elementary Functions. The Elementary Functions as those you’ve learned about already: powers if x including fractional and negative powers (one formula for all), six trigonometric functions, six inverse trigonometric function, exponential functions (one for base e and one for other bases), logarithm functions (natural, and the others).

Formulas are really theorems. The proofs of the formulas are interesting from a mathematical point of view. You should follow along when your teacher shows them to you so that you understand why they work and where they come from. You will not be asked to reproduce the proofs on the AP Calculus Exam.

AP Calculus Course and Exam Description Unit 2 topics 2.5 – 2.7

Why the Derivative?

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You’re now ready to learn about the derivative – one of the two big tools of calculus.

If you graph a function on your calculator and Zoom-In several times at a point on it graph, your graph will eventually look like a straight line. Try it now; pick your favorite function and Zoom in where it looks most curvey. Very close up most functions look like lines. (There are exceptions.) The “slope” of this “line” is the derivative.

A little more precisely, the derivative is the slope of the line tangent to the graph at the point you compute it. It is found by considering a line that intersects the graph at two points and then moving the second point to the first using a limiting process. So, derivatives are always limits.

 The derivative is derived from the function itself. (That’s where the name comes from.)

The difference between the slope of a line and the slope of the curve is this: lines have a constant slope; curves have slopes that change as you move along the graph.

Since the derivative changes as you move along the graph, “derivative” also means the function that gives the derivative (slope) at each point. You will learn how to derive this function from the equation of the curve.

You will often need to write the equations of this tangent line. No, biggie: you have the point and the slope. That’s all you need.

(Hint: Forget slope-intercept! When writing the equation of a line is easiest way is to use the point-slope form, \displaystyle y=f\left( a \right)+{f}'\left( a \right)\left( {x-a} \right) where the point is \displaystyle \left( {a,f\left( a \right)} \right) and the slope is the derivative denoted by \displaystyle {f}'\left( a \right). You only need these three numbers – the two coordinates of a point and the derivative at that point). Drop them into the point-slope equation and you’re done.)

The tangent line is not your geometry teacher’s tangent line. Curves are not circles, so the tangent line may cross the curve at another point, or several other points. Sometimes the tangent line will even cross the curve at the point at the point of tangency!

You will begin by learning how to find the derivative using limits (all derivatives are limits). Then you will learn how to find the derivatives by bypassing the limiting process. That’s a good-news-bad-news thing. The good news is the formulas for finding derivatives make your work very easy and straightforward. The bad news is you’ll have to memorize the formulas. Sorry! Just giving you a heads-up.

Units: Like the slope of a line, the derivative is the instantaneous rate of change of the function at the point you calculate it. Since it is a rate of change, it has rate-of-change units: miles per hour, meters per second, furlongs per fortnight, figs per Newton.

Units are important:

Using the derivative, you will be able to find out useful things about a function. You can find exactly where the function is increasing and decreasing, exactly where it has its extreme values (its maximum and minimum), where it has “problems,” and other things. These in turn lead to practical considerations for the solution of problems in engineering, science, economics, finance, and any field that uses numbers.

Summary: Derivative has several meanings:

  • The slope of the tangent line to the curve, a/k/a “the slope of the curve.”
  • The function that gives the slope at any point.
  • The instantaneous rate of change of the of the dependent variable (y) with respect to the independent variable (x). Therefore, its units are the units of y divided by the units of x.

Derivatives are important and useful. So, let’s drive ahead.

Course and Exam Description Unit 2 topics 2.1 to 2.4

Graph Analysis Questions (Type 3)

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AP  Questions Type 3: Graph Analysis

The long name is “Here’s the graph of the derivative, tell me things about the function.”

Students are given either the equation of the derivative of a function or a graph identified as the derivative of a function with no equation is given. It is not expected that students will write the equation of the function from the graph (although this may be possible); rather, students are expected to determine key features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points. They will be asked to justify their conclusions.

The graph may be given in context and students will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion. See Linear Motion Problems (Type 2)

Less often the function’s graph may be given, and students will be asked about its derivatives.

What students should be able to do:

  • Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
  • Find and justify where the function is increasing or decreasing.
  • Find and justify extreme values (1st and 2nd derivative tests, Closed interval test a/k/a Candidates’ test).
  • Find and justify points of inflection.
  • Find slopes (second derivatives, acceleration) from the graph.
  • Write an equation of a tangent line.
  • Evaluate Riemann sums from geometry of the graph only. This usually involves familiar shapes such as triangles or semicircles.
  • FTC: Evaluate integral from the area of regions on the graph.
  • FTC: The function, g(x), may be defined by an integral where the given graph is the graph of the integrand, f(t), so students should know that if,

\displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{x}{{f\left( t \right)dt}}, then  \displaystyle {g}'\left( x \right)=f\left( x \right)  and  \displaystyle {g}''\left( x \right)={f}'\left( x \right).

In this case, students should write \displaystyle {g}'\left( x \right)=f\left( x \right) on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

There are numerous ideas and concepts that can be tested with this type of question. The type appears on the multiple-choice exams as well as the free-response. Between multiple-choice and free-response this topic may account for 15% or more of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with integrating these topics, so be sure to use as many actual AP Exam questions as possible. Study past exams: look them over and see the different things that can be asked.

The Graph Analysis problem may cover topics primarily from primarily from Unit 4, Unit 5, and Unit 8 of the CED 

For previous posts on this subject see October 15, 17, 19, 24, 26 (my most read post), 2012 and January 25, 28, 2013

Free-response questions:

  • Function given as a graph, questions about its integral (so by FTC the graph is the derivative):  2016 AB 3/BC 3, 2018 AB3
  • Table and graph of function given, questions about related functions: 2017 AB 6,
  • Derivative given as a graph: 2016 AB 3 and 2017 AB 3
  • Information given in a table 2014 AB 5
  • 2021 AB 4 / BC 4
  • 2021 AB 5 (b), (c), (d)
  • 2022 AB3 / BC3 – graph analysis, max/min
  • 2023 AB 4 / BC 4 – graph stem, max/min, concavity, L’Hospital’s Rule

Multiple-choice questions from non-secure exam. Notice the number of questions all from the same year; this is in addition to one free-response question (~25 points on AB and ~23 points on BC out of 108 points total)

  • 2012 AB: 2, 5, 15, 17, 21, 22, 24, 26, 76, 78, 80, 82, 83, 84, 85, 87
  • 2012 BC 3, 11, 12, 15, 12, 18, 21, 76, 78, 80, 81, 84, 88, 89

A good activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Revised March 12, 2021, March 18, 2022, June 4, 2023

Linear Motion (Type 2)

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AP Questions Type 2: Linear Motion

We continue the discussion of the various type questions on the AP Calculus Exams with linear motion questions.

“A particle (or car, person, or bicycle) moves on a number line ….”

These questions may give the position equation, the velocity equation (most often), or the acceleration equation of something that is moving on the x– or y-axis as a function of time, along with an initial condition. The questions ask for information about the motion of the particle: its direction, when it changes direction, its maximum position in one direction (farthest left or right), its speed, etc.

The particle may be a “particle,” a person, car, a rocket, etc.  Particles don’t really move in this way, so the equation or graph should be considered a model. The question is a versatile way to test a variety of calculus concepts since the position, velocity, or acceleration may be given as an equation, a graph, or a table; be sure to use examples of all three forms during the review.

Many of the concepts related to motion problems are the same as those related to function and graph analysis (Type 3). Stress the similarities and show students how the same concepts go by different names. For example, finding when a particle is “farthest right” is the same as finding when a function reaches its “absolute maximum value.” See my post for Motion Problems: Same Thing, Different Context for a list of these corresponding terms. There is usually one free-response question and three or more multiple-choice questions on this topic.

The position, s(t), is a function of time. The relationships are:

  • The velocity is the derivative of the position \displaystyle {s}'\left( t \right)=v\left( t \right).  Velocity has direction (indicated by its sign) and magnitude. Technically, velocity is a vector; the term “vector” will not appear on the AB exam.
  • Speed is the absolute value of velocity; it is a number, not a vector. See my post for Speed.
  • Acceleration is the derivative of velocity and the second derivative of position, \displaystyle {{s}'}'\left( t \right)={v}'\left( t \right)=a\left( t \right) It, too, has direction and magnitude and is a vector.
  • Velocity is the antiderivative of acceleration.
  • Position is the antiderivative of velocity.

What students should be able to do:

  • Understand and use the relationships above.
  • Distinguish between position at some time and the total distance traveled during the time period.
  • The total distance traveled is the definite integral of the speed (absolute value of velocity) \displaystyle \int_{a}^{b}{{\left| {v\left( t \right)} \right|dt}}.
  •  Be sure your students understand the term displacement; it is the net distance traveled or distance between the initial position and the final position. Displacement is the definite integral of the velocity (rate of change): \displaystyle \int_{a}^{b}{{v\left( t \right)dt}}
  • The final position is the initial position plus the displacement (definite integral of the rate of change from x= a to x = t): \displaystyle s\left( t \right)=s\left( a \right)+\int_{a}^{t}{{v\left( x \right)dx}} Notice that this is an accumulation function equation (Type 1).
  • Initial value differential equation problems: given the velocity or acceleration with initial condition(s) find the position or velocity. These are easily handled with the accumulation equation in the bullet above but may also be handled as an initial value problem.
  • Find the speed at a given time. Speed is the absolute value of velocity.
  • Find average speed, velocity, or acceleration
  • Determine if the speed is increasing or decreasing.
    • When the velocity and acceleration have the same sign, the speed increases. When they have different signs, the speed decreases.
    • If the velocity graph is moving away from (towards) the t-axis the speed is increasing (decreasing). See the post on Speed.
    • There is also a worksheet on speed here
    • The analytic approach to speed: A Note on Speed
  • Use a difference quotient to approximate the derivative (velocity or acceleration) from a table. Be sure the work shows a quotient.
  • Riemann sum approximations.
  • Units of measure.
  • Interpret meaning of a derivative or a definite integral in context of the problem

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

This may be an AB or BC question. The BC topic of motion in a plane, (Type 8: parametric equations and vectors) will be discussed in a later post.

The Linear Motion problem may cover topics primarily from primarily from Unit 4, and also from Unit 3, Unit 5, Unit 6, and Unit 8 (for BC) of the CED. 

Free-response examples:

  • 2017 AB 5, Equation stem
  • 2009 AB1/BC1, Graph stem: 
  • 2019 AB2 Table stem 
  •  2021 AB 2 Equation stem
  •  2022 AB6 Equation stem – velocity, acceleration, position, max/min
  • 2023 AB 2 Equation stem – velocity (given), change of direction, acceleration, speeding up or slowing down, position, total distance. 

Multiple-choice examples from non-secure exams:

  • 2012 AB 6, 16, 28, 79, 83, 89
  • 2012 BC 2, 89

Updated: March 15, and May 11, 2022, June 4, 2023

Rate & Accumulation (Type 1)

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The Free-response Questions

There are ten general types of AP Calculus free-response questions. This and the next nine posts will discuss each of them.

NOTE: The numbers I’ve assigned to each type DO NOT correspond to the CED Unit numbers. Many AP Exam questions intentionally have parts from different Units. The CED Unit numbers will be referenced in each post.

AP Questions Type 1: Rate and Accumulation

These questions are often in context with a lot of words describing a situation in which some quantities are changing. There are usually two rates acting in opposite ways (sometimes called an in-out question). Students are asked about the change that the rates produce over a time interval either separately or together.

The rates are often fairly complicated functions. If the question is on the calculator allowed section, students should store the functions in the equation editor of their calculator and use their calculator to do any graphing, integration, or differentiation that may be necessary.

The main idea is that over the time interval [a, b] the integral of a rate of change is the net amount of change

\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)

If the question asks for an amount, look around for a rate to integrate.

The final (accumulated) amount is the initial amount plus the accumulated change:

\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}}

where \displaystyle {{x}_{0}} is the initial time, and \displaystyle f\left( {{{x}_{0}}} \right) is the initial amount. Since this is one of the main interpretations of the definite integral the concept may come up in a variety of situations.

What students should be able to do:

  • Be ready to read and apply; often these problems contain a lot of words which need to be carefully read and understood.
  • Understand the question. It is often not necessary to do as much computation as it seems at first.
  • Recognize that rate = derivative.
  • Recognize a rate from the units given without the words “rate” or “derivative.”
  • Find the change in an amount by integrating the rate. The integral of a rate of change gives the amount of change (FTC):

\displaystyle \int_{a}^{b}{{{f}'\left( t \right)dt}}=f\left( b \right)-f\left( a \right)

  • Find the final amount by adding the initial amount to the amount found by integrating the rate. If \displaystyle {{x}_{0}} is the initial time, and \displaystyle f\left( {{{x}_{0}}} \right)  is the initial amount, then final accumulated amount is

\displaystyle f\left( x \right)=f\left( {{{x}_{0}}} \right)+\int_{{{{x}_{0}}}}^{x}{{{f}'\left( t \right)dt}},

  • Write an integral expression that gives the amount at a general time. BE CAREFUL, the dt must be included in the correct place. Think of the integral sign and the dt as parentheses around the integrand.
  • Find the average value of a function
  • Use FTC to differentiate a function defined by an integral.
  • Explain the meaning of a derivative or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) what the numerical argument means in the context of the question.
  • Explain the meaning of a definite integral or its value in terms of the context of the problem. The explanation should contain (1) what it represents, (2) its units, and (3) how the limits of integration apply in the context of the question.
  • Store functions in their calculator recall them to do computations on their calculator.
  • If the rates are given in a table, be ready to approximate an integral using a Riemann sum or by trapezoids. Also, be ready to approximate a derivative using a quotient from the numbers in the table.
  • Do a max/min or increasing/decreasing analysis.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

The Rate – Accumulation question may cover topics primarily from Unit 4, Unit 5, Unit 6 and Unit 8 of the CED.

Typical free-response examples:

  • 2013 AB1/BC1
  • 2015 AB1/BC1
  • 2018 AB1/BC1
  • 2019 AB1/BC1
  • 2022 AB1/BC1 – includes average value, inc/dec analysis, max/min analysis
  • 2023 AB1/BC1 – Table stem, average value, MVT,
  • One of my favorites Good Question 6 (2002 AB 4)

Typical multiple-choice examples from non-secure exams:

  • 2012 AB 8, 81, 89
  • 2012 BC 8 (same as AB 8)

Updated January 31, 2019, March 12, 2021, March 11, 2022. February 17, 2024