


































v2018i1


Volume 1, Issue 1

Simplifying and Refactoring Introductory Calculus
Jonathan Bartlett

DOI: 10.33014/issn.2640-5652.1.1.bartlett.1

Abstract

First year calculus is often taught in a way that is very
burdensome to the student. Students have to memorize a
diversity of processes for essentially performing the same
task. However, many calculus processes can be simplified
and streamlined so that fewer concepts can provide more
flexibility and capability for first-year students.

1 Introduction

While the exact set of topics in any particular calculus book
or course may vary, the general method of calculus train-
ing has been essentially set in stone for the last hundred
years. Nonetheless, there are many issues with this method-
ology that have been insufficiently addressed over the years.
These issues range from the sequencing of topics to the con-
tent of the topics themselves.

A well-recognized phenomena in computer programming is
known as “code debt.” Code debt occurs when new ideas
and changes get incorporated into a program, but the rest
of the program doesn’t change sufficiently to take these new
ideas and changes into account. Because of this, there winds
up being a lot of duplication and confusion over the right
ways of doing things. It is known as code debt because
eventually, to relieve the tension, future work will have to
be done to the rest of the system to bring it back into
alignment.

To relieve code debt, computer programmers engage in in a
process known as “refactoring.” The idea behind refactoring
is to reconceptualize the whole of the computer program in
order to discover which facilities are the core, distinguish-
able pieces and which ones are merely a variant or permu-
tation of those pieces.

The present goal of the paper is to begin a similar process
of refactoring the subject of calculus and the way that it is
taught.

2 Proof Mechanisms and Learning
Calculus

Calculus books and courses are often written so that each
step of the development of calculus can be proved. Most
books begin with limits, and then calculus is then proved
using limits. Unfortunately, most calculus students are not
yet ready for limits, and limits often wind up being a con-
fusing side-challenge that derail students before they have
even begun to study the subject in depth.

In nearly any subject, students do best when proofs occur
after the core content is learned. As an example, consider
children learning a language. Children learn English (or
any other language) long before they learn the grammar of
English. In fact, most students can speak English perfectly
well without ever learning the rules (grammar) that govern
English. Even when learning foreign languages, immersive
approaches (i.e., those in which the language precedes the
grammar of it) tend to work most effectively (González-
Lloret and Nielson, 2015).

Of course, mathematical learning is not identical to lan-
guage learning. Part of mathematics is proof, so learning
proof tools is certainly an important part of mathemat-
ics. Additionally, with most technical subjects, having a
grounding in the “why” aspects certainly helps one to learn
the intuitions behind the “how.” However, even when the
proofs help build an intuition, there is a difference between
teaching an intuition behind an idea by proving its truth
and trying to get students to construct their own proofs ex
nihilo. Therefore, the sequencing in this method will focus
on first making the intuitive understanding work, and then
only later establish proofs.

The primary target of this reasoning is limits. Most of
the time, limits are introduced before any of the problems
it is intended to solve, which makes the subject confusing
in the first place. If a student has grown up doing the
simple calculation f (3) when f (x) = x2, it makes no sense
to transform this into limit notation, lim

x→3
x2. This is simply

seen by the student as extra steps which have no perceptible
benefit.



18 Simplifying and Refactoring Introductory Calculus

Students see the new notation, a bunch of rules, and are
asked to put them together to solve trivial problems that
they could solve using the normal rules of algebra. There-
fore, they start to get lost because they are bombarded with
extra notation for nothing. They can’t grasp at what this
could possibly be for, and are lost before they ever really
start.

At minimum, the concept of limits should be introduced
to solve the problem of having “holes” in a graph rather
than for expressions that are easily calculated without lim-
its. This at least provides a good context for why students
should bother with them. In my own classes, I have moved
limits to the end of the course altogether.

Some want to introduce limits to establish a formal def-
inition of continuity. However, it is much more straight-
forward to new students to introduce continuity as “being
able to draw a graph without picking up your pencil” than
to talk about limits. Students understand drawing without
picking up your pencil much better than limits. After a
whole year of talking about continuity in terms of whether
or not you pick up your pencil (and why that is important
to calculus), you can say that, as another benefit of limits,
we get a more formal definition of what it means to have a
continuous graph.

Thus, the explicit justification for the derivative using lim-
its occurs long after the students learn the derivative and
learn to use it well. This has the additional advantage of
including L’Hospital’s rule as part of the general discussion
of limit behavior, rather than having to revisit limits again
after derivatives.

3 Introducing the Derivative

In this method, derivatives are introduced through exten-
sive practical work on evaluating slopes between two points
on a graph. Given a graph of, say, y = x2, what is the slope
between the points x = 1 and x = 2? Basic algebra is used
to to find the slope from two points, using the standard
formula

m =
y1 − y0
x1 − x0

.

Next, we discuss what it means to find the slope at a point.
We talk about the problems of finding a slope at a single
point (the formula we have reduces to 0

0 ), and then discuss
methods of getting around that. I suggest estimating the
slope at the point using two points that are close together.
Several problems are worked for getting the distance be-
tween the two x locations closer and closer (x = 1 and

x = 1.1, then move to x = 1 and x = 1.0001).

Eventually, we try to make a formula that expresses the
“general idea” of a slope between two points on a given
line. If we are looking at two points where the x values are
0.01 away from each other, we eventually come up with the
formula m = (x+0.01)2−x2

(x+0.01)−x to match their existing knowledge
of the slope formula.

After solving a few of these, I then suggest that we introduce
a parameter to represent the distance. This way, we can
decide later how close we want the points to be. If we call
our parameter h, then the formula becomes m = (x+h)2−x2

(x+h)−x .
This simplifies to m = 2x + h. So, to find the slope between
any two points that are h = 0.01 away from each other,
we simply use the formula. We then make h smaller and
smaller and smaller to get a “more exact” value of the slope
at a point. I then ask them to think about if there is any
way to abuse this formula in order to find a formula for the
slope at a single point. At this point, students are ready to
understand that if I set h to 0 that the slope is equivalent
to the slope at a particular point.

I then generalize this with functions, using f (x) to represent
the function we want the slope of, and show that basic
simplifications give us the formulat for the derivative,

y′ =
f (x + h) − f (x)

h
.

I then say that, for any particular f (), after getting rid of
h in the denominator through simplification, we can simply
substitute 0 for h in the resulting expression.

I do mention that h is not really zero, but in fact really close
to it. Nonetheless, I tell them that the formal justification
for treating h as zero will come later in the course, and that
for now they should treat it as a value that is sufficiently
close to zero that it can be treated as such, but sufficiently
far away from zero that it doesn’t produce divide-by-zero
errors. Doing this helps students start building intuitions
around these types of situations, which will help us refine
them in our formal discussions of limits later on.

4 Derivatives vs. Differentials

There are two potential operators to use for teaching dif-
ferential calculus. The first, and more commonly used, is
the derivative operator, which can be written as Dx () or as
d
dx (). This is commonly spoken of as “taking the derivative
with respect to x.”

The problem with this operator is that it becomes very



Volume 1, Issue 1

4. DERIVATIVES VS. DIFFERENTIALS 19

confusing for students when they get to derivatives of im-
plicit functions. The reason for this is that, when applied to
implicit functions, derivatives are spitting out other deriva-
tives rather than simple answers. And, in fact, it does this
for some variables and not others. For instance, the deriva-
tive x with respect to x is 1, but the derivative of y with
respect to x is dy

dx . This assymetry causes confusion.

While the derivative operation is perfectly consistent and
solves the necessary problems, a much clearer way to teach
students all types of derivatives is to instead focus on the
differential operator, d(). The differential operator is sim-
ilar to the derivative, but it does not treat any variable
specially. There is no variable that the derivative is being
taken with respect to. Instead, whichever variable(s) are in
the expression always comes out at the end as a differential.

To understand the differences, we will apply the operators
to both an explicit and an implicit function. The explicit
function will be y = x3 and the implicit function will be
xy = 5.

Using the d
dx () operator, the explicit function is straight-

forward.
d
dx

(y) =
d
dx

(x3)

dy
dx
= 3x2.

However, the implicit function is unintuitive for students.

d
dx

(xy) = 5

x
dy
dx
+ y = 0

The product rule, which is supposed to have some amount
of symmetry to it, now seems to have no symmetry what-
soever.

However, when taking differentials, the operations are much
more straightforward. First, the explicit function.

d(y) = d(x3)

dy = 3x2 dx

As you can see, this is identical to the d
dx () operation except

that the result is not divided by dx. This localizes the
differentials to the contexts where they occur. As we will see
shortly, this simplifies a lot of reasoning about complicated
derivatives.

Here is the differentiation of the implicit function.

d(xy) = d(5)
x dy + y dx = 0

This is functionally identical to the implicit derivative.
However, it maintains the symmetry of the product rule
itself. Additionally, it required no extra rules for the stu-
dent for the handling of y vs. x. They are both treated
identically in the differentiation step.

To convert the result of differentiation into a derivative, the
student merely needs to solve for dy

dx using normal algebra
rules. Therefore, all derivatives become a two-step process:

1. Find the differential of the equation.

2. Solve for the derivative you are interested in.

This way, the process for finding dy
dx is the same as for find-

ing dx
dy . There are no funny steps. You merely differentiate

both sides and then solve for the particular derivative (ra-
tio of differentials) that you want. It does not matter if
the equation is implicit, explicit, or, as we will see shortly,
multivariate, the process is identical.

Even though going through the differential is slightly more
complicated for simple equations, overall I have found that
students find the separation of finding differentials and then
finding derivatives to be much easier to process over a wider
variety of problems.

Interestingly, this is the way the original Leibnizian cal-
culus was developed. The process focused on differentials
rather than derivatives, and did not focus on finding ratios
of differentials or worrying about identifying independent
variables (Bos, 1974). As mentioned in Section 3, I do start
with the traditional derivative, as it is much more concrete.
However, I do attempt to pivot very quickly from deriva-
tives to differentials.

Another possible mechanism for doing this was pioneered
by Thomson (1910). In this method, differentials are intro-
duced first, basically as almost-discrete differences. Essen-
tially, the question is asked, given, say, y = x2, if I added
a number named dy to all the y variables, I would have
to compensate by adding some other number dx to all the
x variables. This possibility is also explored, but not very
fully, by Dray and Manogue (2010). Future work may entail
discovering which approach is more beneficial to students—
starting with slopes as a concrete tie-in to algebra and then
pivoting to differentials, or starting more directly with dif-
ferentials at the very beginning.



20 Simplifying and Refactoring Introductory Calculus

5 Treating Differentials
Algebraically

Because we are solving for the derivative instead of calcu-
lating it directly, we must be willing to treat differentials al-
gebraically. Although in the 19th century this was frowned
upon, developments in the 20th century have shown that
differentials can be thought of as algebraic units. Abraham
Robinson’s formal description of the hyperreals and the in-
troduction of non-standard analysis provides sufficient jus-
tification for treating differentials algebraically (Robinson,
1974; Henle and Kleinberg, 2003; Keisler, 2012). Essen-
tially, differentials are treated as hyperreal values—values
which exist on the number line but which are not contained
by the reals.

Although differentials can be treated algebraically, they
can’t actually be solved for as specific numbers. Eventually,
to get a number that can be evaluated or calculated, the
differentials have to be in ratio with each other. However,
for intermediate steps, there is no problem with treating
differentials such as dx and dy exactly as you would the
variables x and y. They can be multiplied, divided, can-
celled, etc. Treating differentials as algebraic units is very
intuitive for students who have studied and practiced the
algebraic treatment of unknowns for quite some time.

Additionally, the ability to treat differentials algebraically
improves the comprehension of topics such as related rate
problems. In related rate problems, dt springs up as if from
nowhere. Books often teach this as “taking the derivative
with respect to t,” but this is confusing since there is no t.
I have found that students are much less confused by an al-
gebraic approach. We simply take the differential the same
way we always have, but then we divide both sides of the
equation by dt. This works simply because, algebraically,
we can divide by any value we wish as long as we divide by
the same thing on both sides of the equation, even if it is a
new unknown.

Again, by using differentials instead of derivatives, we have
transformed a number of processes that students find unin-
tuitive into a single process where the intuition is supplied
by the student’s knowledge of algebra.

6 Multivariable Calculus

This method of using differentials instead of derivatives also
simplifies some aspects of multivariable calculus. Take the
equation z2 = xy. This can be easily converted into differ-

entials.

d(z2) = d(xy)
2z dz = x dy + y dx

This can then be solved algebraically for any derivative that
is desired. For instance, dy

dz can be found.

2z dz = x dy + y dx
2z dz − y dx = x dy
2z
x

dz − y

x
dx = dy

2z
x
− y

x
dx
dz
=

dy
dz

What this means is that the rate at which y changes with re-
spect to z not only depends on the actual values of x, y, and
z, but that it also depends on the rate at which x changes
with respect to z. Many calculus books skip over this idea,
probably because when differentials are not treated inde-
pendently, these processes are very complicated. However,
with dealing with differentials instead of derivatives, these
ideas arise very naturally from the notation.

7 Partial Derivatives/Differentials

This treatment of differentials also leads to a very straight-
forward way of defining partial differentials and derivatives.
Total derivatives define the relationships between changes
in all parts of the system. But what happens if you want
to hold some pieces still, and only find the way that one
variable influences another when treated by itself?

In order to do that, we would say that the other variables
don’t change. Another way of saying that these variables
don’t change is to say that their differentials are zero.

In other words, to convert from a multivariable differential
(as in Section 6) to a partial differential is to merely set the
other differentials which aren’t under consideration to zero.

For instance, Section 6 ended with the equation

dy
dz
=

2z
x
− y

x
dx
dz
.

To understand the relationship between the changes in y
and z if everything else is kept constant, we merely need to
set the differentials of everything else (dx in this case) to



Volume 1, Issue 1

8. HIGHER ORDER DIFFERENTIALS AND DERIVATIVES 21

zero. Doing this results in the equation

dy
dz
=

2z
x
− y

x
dx
dz

∂y

∂z
=

2z
x
− y

x
0
dz

=
2z
x
− 0

=
2z
x

Therefore, the partial derivative of y with respect to z is
2z
x .

Note, however, that partial differentials cannot always be
treated algebraically. This is not due to a failing of the con-
cept of partial derivatives, but merely of their notation. For
our original equation, ∂y can refer to two different entities
in the ratios ∂y

∂z (where dx = 0) and ∂y
∂x (where dy = 0). In-

formation about which particular ∂y is being spoken about
is contained in the denominator, and therefore splitting the
numerator from the denominator results in a loss of infor-
mation.1

8 Higher Order Differentials and
Derivatives

Performing derivatives by taking differentials first also has
benefits down the road, as it allows students to understand
other notations more fully. The Liebniz notation for the
second derivative

!
d2y
dx2

"
has long baffled many students.

Most books take a “just use it and don’t ask questions”
approach.2 However, recent advances have shown that the
standard notation for higher-order derivatives is not only
baffling, it is in a very real sense incorrect (Bartlett and
Khurshudyan, 2018).

To understand the issues, first recognize that differentials
are actually a shorthand. When you take the differential of
a composite function, you always wind up with a differential
of the inner function. For instance,

d(sin(x2)) = cos(x2) d(x2) = cos(x2) 2x dx.
1A possible solution to this problem would be to subscript partial

differentials with the list of differentials which were allowed to change.
This is cumbersome, but allows for an algebraic treatment of partial
differentials. So, for instance, ∂y

∂z would be written as ∂yz y
∂yz z

and ∂y
∂x

would be written as ∂yx y
∂yx x . Doing this makes it clear ∂yz y is a distinct

algebraic entity from ∂yx y. This is the subject of a current paper in
progress from the present author.

2Bartlett and Khurshudyan (2018) gives several examples of text-
books taking this approach.

However, the last term, dx, stands in the same relationship
to 2x as d(x2) does to cos(x2). It is the differential of the
interior function. Therefore, dx actually is just a shorthand
for d(x). Since it is irreducible, it is shortened to just dx.
This shortening happens both to reduce reading and writ-
ing effort, and also to mark the fact that this differential
cannot be further reduced. Nonetheless, dx is actually a
composite—an operator and an operand.3

But what do d2y and dx2 mean? The latter is straight-
forward enough. dx2 is simply a shorthand for (d(x))2.
However, d2y (with the superscript after the differential
operator) actually means applying the differential operator
twice. In other words, while dy is short for d(y), d2y is a
shorthand for d(d(y)).

To see how this plays out, imagine the equation

y = x3. (1)

Let’s start by taking the differential of this equation twice.

y = x3

d(y) = d(x3)

dy = 3x2 dx first differential

d(dy) = d(3x2 dx)

d2y = 3x2 d2x + 6x dx2 second differential (2)

This result may seem surprising, but allow for an explana-
tion. The term 3x2 d2x seems like it is out of place, but it is
not. Since 3x2 dx is the product of 3x2 and dx, the product
rule has to be used to resolve the next differential.

Therefore, since d(uv) = u dv + v du, while one of the out-
puts is the anticipated 6x dx2, there is also another one,
3x2 d2x. If x is the independent variable, this term goes
to zero because d2x goes to zero (see Bartlett and Khur-
shudyan (2018) and Bos (1974) for an explanation of why),
and the result looks like it is expected to under typical
understandings of calculus. However, if x is not an inde-
pendent variable, the term is vitally important. Without
keeping the term, the second differential would not be alge-
braically manipulable. Keeping the term then allows higher
order differentials to remain as algebraically manipulable
terms.

Now, let us divide the whole thing by dx2. Doing so yields

d2y

dx2 = 3x2 d2x
dx2 + 6x (3)

3To emphasize this, I usually typeset dx such that the d is in
roman type and the x is in italic. This is similar to the way that
other functions such as sin x are typeset.



22 Simplifying and Refactoring Introductory Calculus

However, this has more terms than what we normally ex-
pect from a second derivative. This is because the second
derivative comes from the following sequence of steps:

1. Take a differential

2. Divide by dx

3. Take another differential

4. Divide by dx

Because the derivative is the combination of steps 1 & 2,
the second derivative simply repeats those steps. However,
if you take the differential twice in a row before dividing by
dx2, then you wind up with a strange-looking answer, as
we did in Equation 3.

A few things to note about Equation 3:

1. The left-hand side has the form that we normally as-
sociate with the second derivative.

2. The right-hand side does not have the form that we
normally associate with the second derivative. It has
an extra term in it.

3. The reason for this is that, even though it is normally
associated with the second derivative, the left-hand
side is not the correct notation for the derivative of
the derivative of y.

4. In other words, the equation given by Equation 3 is
correct as far as it goes (both sides are in fact equal),
but it is not the second derivative, because the notation
we have come to associate with the second derivative
actually refers to a different quantity altogether.

According to Bartlett and Khurshudyan (2018), the full no-
tation for the second derivatives should be

d2y

dx2 −
dy
dx

d2x
dx2 . (4)

This can be deduced simply from taking two derivatives of
y. The first derivative is obviously dy

dx . The second deriva-
tive is found by taking the differential of dy

dx and then di-
viding by dx.

d
# dy
dx

$

dx
=

dx d(dy)−dy d(dx)
dx2

dx

=
dx d(dy) − dy d(dx)

dx3

=
dx d(dy)

dx3 − dy d(dx)
dx3

=
d(dy)
dx2 −

dy
dx

d(dx)
dx2 (5)

By noting that d(dy) = d(d(y)) = d2y and d(dx) =
d(d(x)) = d2x we can see that this is equivalent with Equa-
tion 4.

So how does this square with Equation 3?

If we subtract 3x2 d2x
dx2 from both sides of Equation 3, it will

result in

d2y

dx2 − 3x2 d2x
dx2 = 6x.

If you recognize 3x2 as being the first derivative (i.e., dy
dx )

it is apparent that the left-hand side of this equation is the
same as the improved form of the second derivative listed
in Equation 4.

Because we are using this form of the second derivative,
this formula can be algebraically rearranged to yield a wide
variety of results. We can rearrange the terms to find the
second derivative of x with respect to y (instead of the sec-
ond derivative of y with respect to x). Or, it can be com-
bined with other formulas and differential formulas (say,
the relationship of x to some variable t) to algebraically ac-
complish a change of variables. Previously, this was only
available using specialty formulas such as Faà di Bruno’s
formula.

Thus, teaching using Leibnizian differentials confers numer-
ous advantages for higher order differentials:

1. The notation is clearer, because there is a definitive
reason for the way that the notation looks.

2. The notation allows differentials to be modified alge-
braically, which was not previously possible for higher
order derivatives.

3. Because the notation is algebraically manipulable, the
notation allows for students to easily find relationships
that previously required memorized formulas.

9 Getting Rid of Logarithmic
Differentiation

Logarithmic differentiation is the process taught by most
calculus textbooks for taking the derivative of functions of
the form uv . Essentially, what is taught is to use logarithms
to remove the exponent, and then take the derivative now
that the exponent is removed. The problem with this is
that it is needlessly complicated, and forces the student



Volume 1, Issue 1

10. HYPERREALS FOR LIMIT ANALYSIS 23

to use different processes depending on the derivative in
question.4

Before learning logarithmic differentiation, the student had
a more-or-less unified process for taking derivatives:

1. Look at the form of the function.

2. Find the corresponding rule (memorized or from the
book).

3. Apply the rule.

However, logarithmic differentiation breaks that process,
adding extra steps in some circumstances. If the mathe-
matics required this, that would be one thing. However,
there is actually a rule available for forms of the type uv

which is rarely mentioned even in the appendices of most
calculus books. The rule is

d(uv ) = vuv−1du + ln(u)uvdv (6)

This formula can be derived in many ways. The most
straightforward is to first set z = uv and then differenti-
ate both sides using logarithmic differentiation (logarithmic
differentiation is useful to prove the formula for uv but af-
ter that it is fairly useless because you can just apply the
formula). Doing this yields

z = uv

ln(z) = ln(uv )
ln(z) = v ln(u)

d(ln(z)) = d(v ln(u))
1
z
dz =

v

u
du + ln(u)dv

dz =
zv
u

du + z ln(u)dv

dz =
v uv

u
du + ln(u) uvdv

dz = v uv−1du + ln(u) uvdv

This is a decent proof for first-year calculus. However, a
more interesting proof can be found by simply recognizing
that a total differential is the sum of its partials. So, the
partial differential of uv with v kept constant is v uv−1du
and the partial differential of uv with u kept constant is
ln(u)uvdv. Therefore, the total differential is merely the
sum of these, as you can see in Equation 6.

Probably due to its rarity of actually appearing in text-
books, this rule has been given a variety of names, the

4The process of logarithmic differentiation is necessary in the proof
of some important rules, but not in their usage.

two most common being the generalized power rule and
the functional power rule.

Logarithmic differentiation does have some unique uses, but
these are mostly upper-level ideas. For instance, logarith-
mic differentiation can be used to convert products of se-
quences (which are difficult to integrate) into a sum of se-
quences (which tend to be easier).

For instance, logarithmic differentiation allows us to say
that

If y =
n%

j=1
f (x, j)

Then y′ = y
n&

j=1
d(ln( f (x, j))) (7)

Logarithmic differentiation is also sometimes used to sim-
plify complicated fractions, exponents, and even products,
but, since there are already rules for all of these, the “sim-
plification” usually just makes learning the process more
difficult for students. Students need unified processes more
than they need tricks to make things easier. Conceptual
simplicity is usually preferable to speed of computation.

10 Hyperreals for Limit Analysis

ϵ-δ proofs have long been the bane of calculus students.
While they do present an interesting mathematical tech-
nique, the retention rate for understanding ϵ-δ proofs is so
low as to hardly be worth doing (Katz and Polev, 2017).
Instead, the method which helps students understand the
process of limits the most is to use the hyperreal number
line.

On the hyperreal number line, ϵ is the unit of the infinites-
imal (not the same as the ϵ in ϵ-δ proofs). To take a right-
handed limit, the student merely replaces x with x + ϵ ev-
erywhere it occurs in the expression. This “looks at” the
function immediately to the right of the point in question—
an infinitely small step to the right. The left-handed limit
is found by subtracting ϵ .

As an example, the expression x2−25
x−5 cannot be evaluated

at x = 5 because it results in a zero in the denominator.
However, any value except 5 will work. Therefore, if we
bump x an infinitely small amount to the right, the divide-
by-zero problem will no longer exist. Therefore, to find



24 Simplifying and Refactoring Introductory Calculus

lim
x→5+

x2−25
x−5 , we merely replace x with x + ϵ . This yields

lim
x→5+

x2 − 25
x − 5

=
(x + ϵ )2 − 25

x + ϵ − 5

=
x2 + 2xϵ + ϵ2 − 25

x + ϵ − 5

=
(5)2 + 2(5)ϵ + ϵ2 − 25

(5) + ϵ − 5

=
25 + 10ϵ + ϵ2 − 25

ϵ
= 10 + ϵ

This is infinitely close to 10. Therefore, the limit is 10.

Additionally, I have found from experience that limits are
best taught at the end of a year of calculus rather than at
the beginning. Teaching about hyperreal numbers and then
using them in limits provides a good way to make many of
the intuitions developed over the first year of calculus more
rigorous.

11 The Integral as an Infinite Sum

One more change that improves calculus for first-year stu-
dents is redefining the integral from an area to an infinite
sum. This is a very subtle difference, but one that I have
found to be important. Finding the area under a curve is
one particular usage of the integral. However, more gener-
ally, the integral is used as a tool of summation.

Using the idea of the integral as a tool of summation helps
explain the usage of the integral as the area under the curve,
but the converse is not true—explaining the integral as the
area under the curve does not help students imagine other
uses of the integral such as arc lengths and volumes of rev-
olution.

When the integral is defined as an infinite sum of infinitely
small pieces, then it is straightforward to then say that what
is to the right of the integral defines what each individual
small piece that we are adding together looks like. Figure 1
lists several different sorts of things we can add together
using integrals. In each of these, the integral represents the
infinite sum of well-defined infinitely small objects. If the
integral is defined as an infinite sum, this makes an intu-
itive connection for the student. However, if the integral is
defined as the area under the curve, jumping out of this to
understand how the area under the curve can be reconfig-
ured as one of the other operations is quite confusing.

Additionally, defining the integral as an infinite sum allows
for the integral to be more easily defined in multivariable

situations. The reason for this is that, as an infinite sum,
the integral is adding up all of the differences that occur
into a total difference. This works just a straightforwardly
for multivariable differentials as it does for single variable
differentials.

For instance, let’s say we have the equation

dz = dy + dx.

The integral of this is just

z = y + x + C.

As an “area under the curve” this makes no sense. However,
as an “infinite sum” this makes perfect sense. If dz repre-
sents the sum of dy + dx, then the total sum will be the
difference of y+ x+C evaluated at two points. So, traveling
from x = 2, y = 3 to x = 7, y = 4 will give a sum total of

(7 + 4 + C) − (2 + 3 + C) = 11 + C − 5 − C = 6.

Therefore, treating the integral as an infinite sum not only
helps students generalize the integral to various integration-
related formulas, but it also helps students generalize the
integral into a multivariable version, and similarly to com-
plex number situations.

The area under the curve can be seen as a particular, easy-
to-understand instance of infinite summation.

One other benefit of treating the integral as an infinite sum
is that it makes the notation more clear. Some introduc-
tory texts, when defining the integral as the area under the
curve, simply use the integral form as a pro forma way of
specifying the variable of integration.

For instance, it will often be explained like this:

' formula to be integrated(!!!!!!!!!!!!!!!!!)*!!!!!!!!!!!!!!!!!+
x3 + 2x2 − 3x + 5 dx*+()

variable of integration

However, doing it this way needlessly restricts the usage of
integration to only area under the curve, and makes the
other uses of it harder to understand.

12 Resistance to Hyperreals

The primary reason that these methods are not more
widespread is the lack of enthusiasm for the hyperreal num-
ber system. This comes from two sources. The first is
a historical bias against infinities and infinitesimals, with



Volume 1, Issue 1

12. RESISTANCE TO HYPERREALS 25

Figure 1: Summations Using Integration

general geometry infinitesimal geometry formula integral
area under the curve ultrathin rectangles height · width

,
y dx

arc length line length
-

(x1 − x0)2 + (y1 − y0)2
, -

dx2 + dy2

revolution about x ultrashort cylinders π r2h
,
π y2 dx

revolution about y ultrathin shells (curved boxes) height · length · thickness
,
y 2πx dx

infinitesimals actually having the stronger of the negative
biases. The primary bias against infinitesimals comes from
the fact that they were used long before they were proved.

Before rigorous means of working with infinitesimals were
established, many critics pointed to the inconsistent ways
in which infinitesimals were handled as proof that they were
non-entities. In some circumstances, infinitesimals were
treated as zero and thrown away, and, in other circum-
stances, they were used in denominators and were there-
fore treated as non-zero entities which could also be used
for algebraic cancelling. In the latter half of the twen-
tieth century, Robinson’s hyperreal number system made
these operations rigorous with the hyperreal number sys-
tem (Robinson, 1974). However, by that time, the damage
from hundreds of years of skepticism had already taken its
toll.

The most classic recrimination against infinitesimals was
done by George Berkeley, in his famouse quote:

And what are these Fluxions? The Velocities of
evanescent Increments? And what are these same
evanescent Increments? They are neither finite
Quantities nor Quantities infinitely small, nor yet
nothing. May we not call them the ghosts of de-
parted quantities? (Berkeley, 1734)5

Additionally, even though the hyperreal number system has
been shown to be usable as a consistent system for the in-
clusion of infinities and infinitesimals, it is not the only vi-
able candidate. Cantor’s transfinite number system (Can-
tor, 1915), surreal numbers (Knuth, 1974), dual numbers
(Wolfe, 2014), and other systems have all been proposed
for extending the real number line into infinities and in-
finitesimals. While the hyperreals are the most widespread,
it is not the only system available. This lack of standard
convention has prevented a lot of building on any one foun-
dation.

5In modern terminology, fluxions refer to derivatives, and evanes-
cent increments refer to infinitesimals.

Finally, there is also a philosophical distrust of hyperreal
numbers based on general concerns over infinities and in-
finitesimals. Both infinities and infinitesimals have troubled
certain schools of mathematicians. These concerns can be
divided into ontological concerns and epistemological con-
cerns.

The ontological concern is about whether infinities exist
and/or are needed in mathematics. This concern stems
from the line of mathematicians following in the footsteps
of Leopold Kronecker. Kronecker opposed Cantor’s theory
of infinities because Kronecker only admitted mathematical
concepts which could be constructed in a finite number of
steps from the natural numbers (Dauben, 1990).

David Hilbert’s program was similar to Kronecker’s. How-
ever, Hilbert was not actively opposed to infinity per se,
but believed that infinities were ideals and the natural num-
bers were reality. Therefore, Hilbert believed that any true
statement about natural numbers that were proved with in-
finity could also be proven without them. Infinities weren’t
invalid, they were just superfluous. Thus, Hilbert’s program
was to define a finite set of axioms which were consistent
and complete, and could prove any valid theorem without
explicitly relying on any concepts of infinity (Zach, 2016).
Gödel later proved that Hilbert’s program was unworkable,
but the motivations and concerns behind it remain today.
Even for mathematicians who accept Cantor and Gödel’s
infinities, a continued concern about over-reliances on in-
finities and infinitesimals remains.

For the epistemological concern, I cannot point to any one
bright source. However, this concern stands as a subtext to
many conversations I have had regarding hyperreal num-
bers, infinities, and infinitesimals. Essentially, the idea is
that since we cannot point to anything that is infinitely
small or infinitely big in the world around us, it is not safe to
make solid assertions about such entities, as those assertions
cannot be tested. Our minds (and therefore our proofs) can
always fail, therefore, building on premises which can only
be proved logically and not physically is dangerous.



26 Simplifying and Refactoring Introductory Calculus

While these concerns are understandable, the long-term
payoff will come from helping students see the concepts
of the infinite and infinitesimal more clearly, and that will
only come from practice and familiarity with the concepts.
The infinite should not be feared, in fact, mathematics is
one area which allows us to grapple with the infinite on a
much more rigorous basis.

13 Conclusion

By making a few modifications to the way that calculus is
taught, students can be presented a more unified, holistic
system. This is both easier to use and easier to under-
stand. Having a single process and expanding out its usage
to more and more complex cases is much more straightfor-
ward than having to reinvent the system at every step, and
forcing students to memorize different processes for differ-
ent situations.

The changes proposed here include:

• Separating differentiation from finding the
derivative. This allows the unification of explicit dif-
ferentiation, implicit differentiation, and multivariable
differentiation. Additionally, it helps explain (and cor-
rect) the notions of higher-order differentials.

• Treating differentials algebraically. This is an ex-
tension of the separation of finding differentials and
derivatives. Additionally, when done correctly, it im-
proves the usability of higher-order differentials and
derivatives.

• Using a rule for uv instead of logarithmic dif-
ferentiation. Instead of forcing students to use dif-
ferent types of processes for different forms, teaching
the generalized power rule allows students to take the
differential of uv directly, just as with every other form.

• Using hyperreals for limits. Hyperreals allow for a
more intuitive approach to limits. Additionally, mov-
ing limits to the end of a first-year course allows stu-
dents to develop intuitions around the derivative first
before seeing the formal proof of their validity.

• Treating integrals as infinite sums. Treating the
integral as an infinite sum instead of an area under
a curve allows for easier generalization of the concept
of the integral into various geometric situations (and
even non-geometric situations). Additionally, this al-
lows for a more straightforward generalization of inte-
gration into the inclusion of multiple variables.

The goal is to simplify introductory calculus while simulta-
neously making it more powerful. This is accomplished by
(using computer science terminology) “refactoring” calculus
into pieces that are more easily recombined, adapted, and
applied to various situations. Since differentials can always
be transformed into derivatives by algebraic rearrangement,
nothing is lost in their treatment as individuated entities.

These ideas have been incorporated into a new text on cal-
culus (Bartlett, 2018), and future study is needed to fully
assess the impact of these ideas (both positive and negative)
on the teaching of calculus.

References

Bartlett, J (2018). Calculus from the Ground Up. Broken
Arrow, OK: BP Learning.

Bartlett, J and A Z Khurshudyan (2018). “Extending the
Algebraic Manipulability of Differentials”. In: arXiv
1801.09553. In Review. url: https://arxiv.org/
abs/1801.09553.

Berkeley, G (1734). The Analyst.
Bos, H J M (1974). “Differentials, Higher-Order Differen-

tials, and the Derivative in the Leibnizian Calculus”.
In: Archive for History of Exact Sciences 14.1, pp. 1–
90. url: https://link.springer.com/article/10.
1007/BF00327456.

Cantor, G (1915). Contributions to the Founding of the
Theory of Transfinite Numbers. Trans. by P E B Jour-
dain. Dover Publications. url: https://www.maths.
ed.ac.uk/~v1ranick/papers/cantor1.pdf.

Dauben, J W (1990). Georg Cantor: His Mathematics and
Philosophy of the Infinite. Princeton University Press.

Dray, T and C A Manogue (2010). “Putting Differentials
Back into Calculus”. In: The College Mathematics Jour-
nal 41.2, pp. 90–100.

González-Lloret, M and K B Nielson (2015). “Evaluating
TBLT: The Case of a Task-Based Spanish Program”.
In: Language Teaching Research 19.5. url: https://
www.academia.edu/6763175/Evaluating_TBLT_The_
case_of_a_task-based_Spanish_program.

Henle, J M and E M Kleinberg (2003). Infinitesimal Calcu-
lus. Dover Publications.

Katz, M and L Polev (2017). “From Pythagoreans and
Weierstrassians to True Infinitesimal Calculus”. In:
Journal of Humanistic Mathematics 7.1. url: http:
//scholarship.claremont.edu/cgi/viewcontent.
cgi?article=1368%5C&context=jhm.

Keisler, H J (2012). Elementary Calculus: An Infinitesimal
Approach. Dover Publications.



Volume 1, Issue 1

REFERENCES 27

Knuth, D (1974). Surreal Numbers. Addison-Wesley Profes-
sional.

Robinson, A (1974). Non-standard Analysis. North-Holland
Publishing Company.

Thomson, S P (1910). Calculus Made Easy. New York:
MacMillan Company. url: http://www.gutenberg.
org/ebooks/33283.

Wolfe, Alan (2014). “Dual Numbers and Automatic Differ-
entiation”. In: The Blog at the Bottom of the Sea. url:
https://blog.demofox.org/2014/12/30/dual-
numbers-automatic-differentiation/.

Zach, R (2016). “Hilbert’s Program”. In: The Stanford En-
cyclopedia of Philosophy. Ed. by Edward N. Zalta.
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