Calculus Without Limits: A Visual Approach to the Rules of...

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Proceedings of the 2016 Conference for Industry and Education Collaboration

Copyright ©2016, American Society for Engineering Education

Calculus Without Limits:

A Visual Approach to the Rules of Differentiation

Andrew Grossfield, Ph.D., P.E.

Vaughn College of Technology

Abstract

The excitement that many of our college age youth find in robotics and drones has led them to

consider careers in engineering and technology. These students may however consider alternate

careers when they confront the definitions, theorems and proofs concerning limits that are in

conventional college math courses. In order to retain these students as educated engineers and

technological specialists, we should consider introducing calculus as the study of classes and

combinations of basic well-behaved curves. The term well-behaved will be used to describe

curves and functions which are continuous and possess unique tangent lines at all points except

at isolated singularities and which do not wiggle excessively.

In this context the visualizations of the curves have immediate bearing on the interests of the

students. The students then have time to develop analytic proficiency and trust in numerical

analysis as it applies to curves. At a later time, these students could be introduced to the concepts

of numerical limiting processes.

Many of the well-behaved curves of calculus were studied in the 18th century before the

formalization of the epsilon-delta definition by Bolzano in 1817. It is perhaps beneficial for

students of the technologies to become adept at 18th century mathematics before embarking on

the wonderful advances of the 19th century mathematics which culminated in the theory of

Lebesgue integration.

When only the well-behaved functions that were known in the 18th century are under

consideration, it is not necessary to define the derivative as the limit of the difference quotient.

Say a function, y = f(x) describes a curve. The derivative of the function at a fixed point on that

curve is defined as the slope of the tangent line at that point. The derivative, y’, thus provides a

measure of the direction of the curve which is described by the function. With this definition of

the derivative, it is possible to provide explanations for the various rules of differentiation that a

beginning engineering student will find reasonable. It is pedagogically unsound to define a

mathematical concept as something that is the result of a process.

Once a student accepts and routinely uses the rules of differentiation he or she will understand

that the same rules might apply to more wild functions and the student will be better situated to

appreciate why the delta-epsilon arguments of Bolzano and Cauchy are needed to treat these wild

functions.

The goal of this paper is to bypass the difficult formal derivations conventionally presented in

calculus courses in order to place the student in a position where he can apply the rules and

follow the analysis that subsequently he will see in his engineering and technology courses.

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Functions, signals and curves are important technological concepts and a student who

understands how calculus bears on these concepts should require no other application.

A brief history of the concepts of calculus

René Descartes in the early 1600’s is credited with the Cartesian plane which provided the basis

of analytic geometry, which enabled the interpretation of algebraic equations in two variables as

curves and vice versa. In the later 1600’s, Leibniz, who was developing calculus, used the word

function. By 1720 Johann Bernoulli was considering functions as algebraic expressions in the

variables x as the horizontal variable and y which was plotted vertically. Soon after Euler used

the notation f(x) for functions and masterfully explored algebraic manipulations of series forms

of functions.

In the early 1800’s Fourier published his studies of series forms, which revealed that there was

much to do if we were going to understand and trust the manipulations of series. It would take

the next century before Henri Lebesgue would complete the effort with his wonderfully rigorous

and non-intuitive theory of integration and Lebesgue measure.

These developments in mathematical analysis during the 19th century were based on Bolzano’s

and Cauchy’s epsilon-delta theory of convergence of sequences. Later mathematicians were

proud of these magnificent achievements, and after sputnik the first math reform permitted the

wording, definitions and proofs to permeate mathematics pedagogy to the detriment of the more

accessible concepts of analytic geometry and the ideas of calculus that prevailed before the

1800’s.

No one denies the validity, achievement and value of modern mathematical analysis. The

question is whether or not it should be introduced first to a population that is not familiar,

comfortable and adept with either high school algebra or with the 18th century approach to curves

and analytic geometry. The outstanding mathematician, Charles Hermite is noted to have written

in 1893, "I turn with terror and horror from this lamentable scourge of continuous functions with

no derivatives.” Is it sensible to provide beginning engineering and technology students with

definitions and concepts purposely constructed to confront that “lamentable scourge?”

Number operations and function operations

Strictly, number operations and combinations of number operations are not functions. A number

operation performs a procedure either on a single number or a pair of numbers in order to

produce another number. But common usage identifies as functions the combinations of

operations on a variable with the corresponding values of another variable. So squaring and

taking the reciprocal of numbers are not functions but y = x2 and y = 1/x are functions. In the

case y = (sin x) 2, squaring is now considered as a function operation. Here, squaring operates on

the curve y = sin(x) to produce a new curve, y = sin2(x). We must note that function operations

are usually not commutative. Interchanging the order produces different results. The curve of

y = (sin x) 2 is not the same curve as y = sin(x 2). As an example it can be seen in figure 1 below

that the curve, y = (sin x) 2 is periodic and never negative, while in figure 2 it is observed the

curve, y = sin(x 2) is not periodic and alternates between plus and minus one.

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Figure 1 y = {Sin(x)}2 Figure 2 y = Sin(x2)

Differentiation and integration are function operations. Differentiation, dy

dx, starts with a well-

behaved curve, y =f(x), and produces another curve y = f’(x) which provides the direction of the

tangent line, m(x), at each point x. Integration starts with a positive, well-behaved curve,

y =f(x), and produces another curve which relates to the area between the original curve,

y = f(x), and an interval on the x-axis.

The structure of differential calculus

Figure 3 Diagram of the structure of differential calculus

The curves, as studied in a calculus course, are described by algebraic equations, which are

commonly derived from physical laws. These equations are presented in varying forms,

composed of the arithmetic operations, on a variety of basic functions. And so it is meaningful to

present the rules organized by the forms, operations, and kinds of functions appearing in the

equations. Below table 1 displays a list of the rules and their corresponding equations in this

paper. Given the equation for a curve, the rules predict at a given point, whether a curve is

heading up or down and how fast and whether a curve is turning up or down and how fast.

The rule for differentiating an implicit form was omitted because it is written using the

multivariable partial derivative notation, which should appropriately be introduced with more

explanation elsewhere. The students should not have to memorize delta-epsilon proofs but they

should see how the rules correspond to curve properties. The students will still need to practice

the parsing of the equations in order to apply the rules in the proper sequence.

Curves /

Functions

Kinds Properties Forms Operations Objectives

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Figure 3 displays the basic components of the study of curves and displays an outline of this

paper. Of course, calculus should include the rules for differentiating the various kinds and forms

of the curves and also the rules for the differentiating the various ways of generating more

complicated curves by combining simple curves. The location of these rules in this paper can be

found in table 1.

Rules for the derivatives of operations on functions Equation

Linearity y = a*f(x) ± b*g(x) 6

Product Rule y = u(x)*v(x) 10

Quotient Rule y = u(x)/v(x) 11

Differentiating a Reciprocal y = 1/v(x) 12

Rules for the derivatives of kinds of functions Equation

Power rule for positive integer exponents y = xn 7

Power rule for negative integer exponents y = x-n 13

Power rule for fractional exponents y(x) = x 𝑞

𝑝 19

The derivative of the function y = sin(x) 24

The derivative of the function y = cos(x) 25

The derivative of the function y = arc sin(x) 31

The derivative of the function y = arc cos(x) 32

The derivative of the function y = ex 33

The derivative of the function y = ln(x) 34

Rules for the derivatives of forms of functions Equation

The chain rule for the chain form y = g(u) ; u = h(x) 15

The implicit, Cartesian forms of algebraic curves

The derivative of the inverse function x = g(y) 18

The derivative of the parametric form y = f(s)

x = g(s) 21

Table 1 The important rules of differentiating and their location in this paper.

Local properties of functions

The following simple properties should be noted. A constant function, y = c whose graph is a

horizontal line, has a slope, m = 0 everywhere. The derivative of a constant is zero. A function

which is a tilted straight line, y = mx + b has a constant direction. The derivative of this linear

function is the constant value, m. At every point the tangent line will maintain an angle with the

horizontal of θ = arctan(m). See figures 4 and 5 shown below.

When m is positive the line rises toward the upper right. When m is negative the line descends

toward the lower right. It should be clear that a tilted straight line cannot have a maximum or

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minimum on the interior of any interval. The maximum or minimum of any tilted straight line

will be on the boundary of any interval which contains its endpoints.

Figure 4 Angle of inclination and deltas Figure 5 A rising, a constant and a

θ = arctan(m); m = tan(θ) falling line

Some curves like the exponential y = ex or an odd cubic y = x

3 + x are continually rising and at

every point on the curve the slope of the tangent line is positive. Such curves are described as

rising monotonically and similarly continually falling curves have negative slopes and are said to

be monotonically descending. The parabola shown below in figure 6 rises in the first quadrant.

Figure 6 The parabola y = x2 in the first quadrant

If the derivative of a function is defined as the slope of the tangent line, then the following visual

properties of the derivative can be noted: As shown in figure 6 at a point where a smooth

function is rising, the slope, m, of the tangent line is positive. If the curve is descending, the

slope is negative. At the extreme points, the maxima and minima, the tangent line is horizontal

and therefore the slope and the derivative are zero as shown below in figure 7.

rise = (y2 - y1) = delta y

run = (x2 - x1) = delta x

angle of inclination = arctan(m)

P2

P1

m = rise / run = ( y2 -y1) / (x2 - x1)

m = delta y / delta x

m = Tan (angle of inclination)y =.5x +2

y = -.5x + 1

y = 1.2

y = x2

y = 2x -1

Rising curve - The tangent line has a positive slope

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Figure 7 Extreme points and Figure 8 The local behavior in

the point of inflection the quadrants of a circle

The following visual properties of the second derivative y'' of a smooth curve must also be

examined: In the case where the direction of the curve is not constant, then the slope function

y' = m(x) is also not constant. In that case at ordinary points the curve either turns up at the point

of tangency away from the tangent line or, it turns down. ‘Turning up’ means that the curve

approaches the tangent line from above and leaves the point of tangency above the tangent line

as in figure 6. In this case the slopes around the point of tangency are increasing. Increasing

slopes means that the derivatives of the slopes are increasing. A description of the position,

direction and turning of the points in the quadrants of a circle is shown above in figure 8.

Summarizing:

1) If the function values are positive, the points of the curve are above the x- axis.

2) If the derivative values are positive, the curve is rising. And

3) if the second derivative values are positive, the curve is turning up.

If the second derivative is negative then the curve is turning down, meaning that the curve

approaches and leaves the point of tangency below the tangent line. On intervals where the curve

is smooth, the curve turns down in a smaller interval containing a maxima, and turns up near a

minima.

The well behaved curves studied in a calculus are mostly continuous and smooth (that is, the

special points: discontinuities, cusps, zeros, extremes and points of inflection, if there are any,

are separate.) The functions of calculus may not be monotonic but may meander, wiggle or

snake. If the functions are continuous and smooth then they will be monotonic on the intervals

bounded by successive high and lows.

If a curve is smooth but not a straight line, then between a minimum and the subsequent

maximum there will be at least one point of steepest ascent. At these points the slopes will be

extreme and the derivative of the slope will be zero; that is, the second derivative of the function

At extreme points the tangent lines are horizontal

At extreme points the tangent lines are horizontal

P1

P2

P3 (5/3, 47/21 )

a maximum

a minimum

an inflection point

x

y

y'' is positive

y is positive

The curve is rising

y' is positive

y'' is negativey'' is negative

The curve is falling

y'' is positive

y' is positive

y is negative

y' is negative

y is positive

y' is negative

y is negative

and turning downand turning down

The curve is risingThe curve is falling

and turning up.and turning up.

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describing the curve will be zero. These special points are called points of inflection, and they

locate when the curve stops turning up and starts turning down. In addition, as shown in figure 7

above, at a point of inflection the curve crosses the tangent line. Also the phenomenon is shown

on page 5 of the paper Visual Differential Calculus14. Corresponding behavior prevails when a

curve descends from a maximum to a minimum.

Examine the function y = .18sin(5x) + x3 – x – 3, shown below in figure 9 together with its

derivative function y = .9cos(5x) +3x2 – 1. The values of y are seen to rise to the left of A and

also to the right of B. The curve is seen to wiggle as it descends from A to B. The derivative

function is seen to be positive where the graph of y is rising to the left of A and to the right of B.

Between the horizontal values of A and B, where y is descending the derivative function is

negative. At the extreme points of y the derivative is seen to have the value 0. The three points of

inflection lay directly below the extreme points of the derivative function.

Figure 9 The function y = .18sin(5x) + x3 – x – 3

Simple vertical manipulations on single valued branches of curves and their interpretation

in algebra: raising, lowering, scaling and flipping

The simple vertical curve manipulations involving the curve and constants are 1) raising and

lowering the curve without altering the shape of the curve; 2) vertically stretching and

contracting the curve and 3) reflecting (flipping) the curve about the x-axis. First, to raise or

lower a curve, add a constant to the explicit form of the curve’s equation. Adding a positive

constant raises the curve, while adding a negative constant lowers it. Raising or lowering a curve

does not affect the direction of the tangent line and therefore does not change the derivative as

shown below in figure 10. Additionally raising or lowering a curve does not affect the shape of

the curve or the rate of turning, and therefore does not change the second derivative. At a fixed

value of x, when a curve is raised or lowered, the corresponding tangent lines will be parallel.

x

y

y = .18sin(5x) + xxx - x - 3

y = .9cos(5x) + 3xx - 1

A

B

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d

dx{y(x) ± c} =

d

dx y(x) Equation 1

Fig 10 Adding a constant Fig. 11 Doubling a curve

A curve can be vertically stretched by multiplying the explicit form by a positive constant which

is larger than one (See figure 11). To compress a curve vertically, multiply the explicit form by a

positive constant whose value is between 0 and 1. Vertically stretching and compressing a curve

scales the vertical differentials proportionately and therefore scales the slopes of the tangent lines

proportionately.

d

dx{c y(x)} = c

dy

dx Equation 2

To flip a curve vertically, multiply the explicit algebraic form of the curve by minus one.

Flipping a curve reverses the vertical differentials and therefore multiplies the derivative by

minus one (See figure 12).

d

dx{– y(x)} = –

dy

dx Equation 3

Figure 12 Flipping a curve

Adding a constant does not affect the

y = 4x - x

slope of the lines at a given horizontal value

y = 4x - x + 32

2y = 2x + 1

y = 2x + 4

y = -2x + 9

y = -2x + 12

Doubling the vertical values

doubles the slope

of a curve

flipping the vertical values

flips the slopes

of a curve

y = 4x -x

y = -(4x -x )2

2

y = 9 - 2x

y = 2x - 9

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Since the branches of curves are single valued and the arithmetic operations are single valued,

performing any of the four operations on two single valued branches, yields a unique resulting

branch. ( The reason mathematicians define functions as single-valued, even though curves in

general are multi-valued, is that single valued operations on single-valued branches produce

single valued branches. ) We must note that adding, subtracting, multiplying and dividing

continuous curves will result in a continuous curve except at the points where the denominator is

zero. Additionally, we must note that adding, subtracting, multiplying and dividing smooth

curves will result in a smooth curve except at points where the denominator is zero.

When smooth curves are added or subtracted vertically, their tangent lines will also be added or

subtracted and the slope of the sum/difference is the sum/difference of the slopes of the original

tangent lines. The effect of graphical addition and subtraction of curves can be observed in

figures 13 and 14 and in the paper12 on visual analysis. We see that adding an oscillation to an

upward drift produces an oscillation that drifts upwards. This leads to the following two rules:

Fig. 13 Adding two curves; Fig. 14 Subtracting two curves;

y=x and y = sin(x) y=x and y = sin(x)

a) The derivative of the sum of two functions equals the sum of the individual derivatives as

shown in figure 13, that is,

{f(x) + g(x)}' = f'(x) + g'(x). Equation 4

b) The derivative of the difference of two functions equals the difference of the individual

derivatives as shown in figure 14, that is,

{f(x) – g(x)}' = f'(x) – g'(x). Equation 5

Any combination of operations of scaling, adding and subtracting functions as in:

a*f(x) ± b*g(x) ± c*h(x),

is called a linear combination of the functions; f, g and h. The above rules can all be combined

into one important statement which states that the derivative of a linear combination of functions

is the linear combination of the individual derivatives. Because the process of differentiation has

this property, it is called a linear operation.

y = x + sin(5x))

The slope of the sum of two curves

is the sum of teh individual slopes.

y = x - sin(5x))

The slope of the difference of two curves

is the difference in the individual slopes.

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d

dx {a*f(x) ± b*g(x)} = a*

d

dx f(x) ± b*

d

dx g(x) Equation 6

Equation 6 states that differentiation and linear combination as function operations are

commutative. Commutativity means that the order in which two operations are performed does

not affect the result. Unfortunately multiplication and differentiation of functions are not

commutative.

In order to continue our study of the rules of differentiation, we will need to review a class of

functions on which we can test the differentiation rules. The functions in the class are simple but

will enable us to construct the polynomials, rational functions and some algebraic functions by

scaling, adding, subtracting, multiplying, dividing and taking inverses. These functions are

called the power functions and are simply the powers of x: y = xn, where n is called the degree

of the power function

Power functions

The power functions are defined for all values of x and are continuous and smooth. If the degree

is odd these functions have odd symmetry and have even symmetry for an even degree. If the

function is of odd degree and has a positive coefficient, c, the functions start in the third quadrant

at ( – ∞ ,– ∞ ) and rise monotonically, continuously and smoothly to the origin and continue

rising into the first quadrant smoothly to + ∞. See figure 15. If c is positive and the degree is

even as shown in figure 16, the power functions start in the 2nd quadrant at ( – ∞, + ∞), descend

continuously and smoothly toward the origin and then turn up and rise smoothly toward

(+ ∞, + ∞) in the first quadrant. The curves of the scaled power functions of 2nd degree, y = cx2

are vertical parabolas. We will now see that the derivative of the power function y = xn can be

shown to be another power function whose degree is diminished by 1, y’ = n x(n-1).

Figure 15 An odd degree power function Figure 16 An even degree power function

y = x + sin(5x))

The odd degree power function

y = x 3

The even degree power function

y = x 2

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A derivation of the rule for differentiating the power functions, y = xn

For a fixed n and a fixed point, P( a , an ) on the curve of the power function, the following is a

polynomial conditional equation relating how the slope of a secant line through P and another

point Q( x , xn ) varies with x.

Δy = xn ─ an = m (x ─ a) = m Δx

Factoring xn ─ an ,

Δy = xn ─ an = (x ─ a) {x (n-1) + x(n-2) a + x(n-3) a2 + x(n-4) a3 + x a(n -2) + a(n -1) }

= m (x ─ a) = m Δx

At the point of tangency where x = a, the slope, m, of the tangent line is the coefficient Δx .

m = (x (n-1) + x(n-2) a + x(n-3) a2 + x(n-4) a3 + x a(n -2) + a(n -1)

This expression has n terms, each of which, at x = a, has the form a(n-1). The derivative of the

power function which is the value of the slope of the tangent line at the point x = a is seen to be:

d

dx an = m(a) = n a(n – 1)

For general values of the horizontal variable, x, the direction of the tangent line can be obtained

from equation 7 which provides the rule for the derivative of a power function:

m =d

dx(xn) = n x(n−1) Equation 7

The rule for differentiating the product of two functions, y = u(x)*v(x)

We are now in a position to see that the derivative of a product will not be the product of the

individual derivatives. Let us try to apply Equation 7 to the product of two power functions. Start

with two power functions, say u = x² and v = x³ , then let y(x) = the product of u and v.

y(x) =u*v = x5 . We find that dy

dx = 5x4 ; but

du

dx = 2x and

dv

dx = 3x² .

And we see that d(uv)

dx =

d( x5)

dx = 5x4 ≠ 6x³ = (2x)(3x²)=

du

dx∗

dv

dx .

It turns out that while the derivative of a product is not simply the product of the derivatives, the

appropriate rule can be found and it is not very complicated. Let u(x) and v(x) represent the

length and width of a rectangle. Then y(x) = u*v will represent the area of the rectangle. See

figure 18. We observe that if u is increased by an amount Δu, and v is increased by an amount

Δv, then the area y is increased to an amount (u + Δu)( v + Δv) = uv + u Δv +v Δu + Δu Δv. It

follows that the increment in area

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Δy = (u + Δu)( v + Δv) ─ uv = u Δv +v Δu + Δu Δv Equation 8

If y and u and v are linear functions of x then Δy = my Δx , Δu = mu Δx and Δv = mv Δx

where my is the derivative of y with respect to x, mu is the derivative of u with respect to x and

mv is the derivative of v with respect to x. Then equation 8 becomes

Δy = my Δx = u*mv Δx + v*mu Δx + mu mv Δx Δx. Equation 9

Equation 9 shows that Δy is related by a quadratic to Δx. But on the tangent line, Δy is linearly

related to Δx and the quadratic term can be discarded. The remaining linear relationship is:

Δy = my Δx = {u*mv Δx + v*mu} Δx.

The derivative of y with respect to x, my, which is the coefficient of Δx is found to be

my = u*mv + v*mu.

In figure 17 the area represented by the non-linear term Δu Δv is seen to be small. The rule,

which is called the product rule, for differentiating a product, y(x) = u(x)*v(x), of any smooth

functions u(x) and v(x) is found to be

dy

dx = u

dv

dx + v

du

dx . Equation 10

∆v u*∆v ∆u*∆v

v(x) u(x)*v(x) v*∆u

u(x) ∆u

Figure 17 Diagram of the Product Rule

Deriving the quotient rule, the reciprocal rule and differentiating a negative power of x

Ordinary algebra can be applied to the product rule to derive a rule for the quotient of functions.

From the explicit quotient form y(x) = u(x)

v(x) , it follows that y(x)* v(x) = u(x) to which the

product rule can be applied to obtain:

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y(x)* dv

dx + v(x) *

dy

dx =

du

dx.

Now we can solve for dy

dx ,

dy

dx=

1

v(x) {

du

dx− y(x) ∗

dv

dx } =

1

v(x)

du

dx−

u(x)

v(x)∗

1

v(x)∗

dv

dx

and then arrive at the quotient rule d

dx{

u

v } =

v du

dx−u

dv

dx

v2 . Equation 11

In the special case if the numerator, u, has the constant value, 1, then the derivative of the

reciprocal of v(x) will be seen to be the reciprocal rule,

𝑑

𝑑𝑥{

1

v } = −

(𝑑v

𝑑𝑥)

v2 Equation 12

One might wonder why there is a negative sign in this equation. If the function v is increasing,

then the reciprocal will be decreasing, and the slope of the tangent line will be opposite to the

slope of the original function, v(x).

If the function is a negative power of x, y = x-n = 1

𝑥n , then the reciprocal rule yields

dy

dx=

d

dx{

1

xn} =d

dx{x−n} = −

nx(n−1)

x2n = − nx(−n−1) Equation 13

Notice that, here too, the power function rule (multiply by the exponent and reduce the exponent

by 1) prevails for all integer exponents.

Derivatives of polynomials and rational functions

The polynomial functions are a class of functions that can be expressed in explicit form as a

linear combination of positive exponent power functions. In this case, the derivative of a linear

combination of power functions is the linear combination of the derivatives of the individual

power functions in the explicit form. For example, to obtain the derivative of the polynomial

function shown in figure 18: y(x) = x3 + x2 + 2x + 2 just add the derivatives of each term. We

observe that the polynomial is always rising and that the derivative is always positive.

dy

dx = 3x2+ 2x +2

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Figure 18 The cubic monotonic polynomial, Figure 19 An asymmetric quartic

y(x) = x3 + x2 + 2x + 2 y(x) = x4 ─ 2x2 + 2x + 1

If the same polynomial is written as a product y = (x2 + 2)(x + 1), the product rule can be

invoked to obtain the same result.

dy

dx = (x2 + 2)(1) + (2x)(x + 1) = 3x2 + 2x +2

Another example of a polynomial is the asymmetrical quartic shown in figure 19.

Rational functions are quotients of polynomials. Examples of the graphs of rational functions

can be found in the paper 12. If the functions are described as quotients of polynomials in

expanded form, the derivative can be computed by invoking the quotient rule.

There is a form for rational functions called “improper”, whose description as an explicit

quotient form has numerator polynomial degree equal to or exceeding the denominator degree.

The improper form can be decomposed into a polynomial plus a simpler form called “proper”

whose numerator degree is less than the denominator degree. Take as an example, the improper

rational function shown in figure 20:

y(x) = x3−2

x−1= x2 + x + 1 −

1

x−1 Equation 14

x

y

y = x + x +2x +223

2y = 3x +2x+2

x

y

A

B

C

D

E

F

G

y = x - 2x + .2x + 124

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Figure 20 An improper rational function and its polynomial part

Because the proper rational functions have a larger denominator degree than their numerator

degree, their values must converge to the horizontal axis as x extends to either + or − ∞. Any

vertical asymptotes would be caused by the zeros in the denominator of this “proper” rational

function. Of course the polynomial part is smooth and the functional behavior as x extends to

either + or − ∞ is determined by sign of the leading term and the degree of the polynomial. It is

easier to differentiate the polynomial and the proper rational function forms than the original

“improper” rational function form. The derivative of the function, y(x), in equation 14 is:

y’ = 2x + 1 + (1

x−1)

2

Differentiating the composite forms of function; the chain rule

Functions are frequently constructed by performing functional operations on other functions. The

Pythagorean trig identity requires squaring both the sine function and the cosine function. The

normal distribution of a random variable requires taking the reciprocal of the number, e, raised to

the power of x2. This concatenation of a sequence of functions is called a composition12 or a

chain.

The composition of the square of the sine might be described by the two equations y = u2;

u = sin x. The variable, x determines the value of u which in turn determines the value of y. The

entire chain involving three variables could, by substitution, be collapsed into the single equation

in two variables y = (sin x) 2. The variable, x, now directly controls the value of the variable y.

x

y

y = x +x+1

An improper rational function

and its polynomial part

2

y =x -1

x - 23

1y = x + x +1 -

2

x - 1

------

------

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For convenience we will refer to sin x as the inner function and squaring as the outer function.

Which form is preferable; two simple equations in three variables or one more complex equation

in two variables? There are times when it is preferable to study the chain of simple operations

and other times when the single more complicated equation is preferred.

Let us examine the case when both the inner and outer operations are linear functions and

consider y and u when y(u) = mu u + bu and u(x) = mx x + bx . Then

y{u(x)} = mu (mx x + bx)+ bu . The derivative of y with respect to x is seen to be the product, mu

mx , of the slopes of the two straight lines. In the equation for u(x), variations in x, which are

magnified by mx , produce corresponding variations in u. In the equation for y(u), changes in u

are magnified by mu and produce changes in y. The effect of the two equations is that changes in

x are magnified by the product of magnifications. When y(u) and u(x) are curves, the derivatives

of the functions will be the slopes of the tangent lines and the composite derivative will be the

product of component derivatives:

dy

dx=

dy

du∗

du

dx Equation 15

This is the rule, called the chain rule, for computing the derivative of a function constructed by a

composition of functions. Leibniz is usually credited with the marvelous notation that in the

chain rule allows us to treat the differential ratios, suggested by the notation, as ordinary

fractions. We should note the wonderful conclusion that whenever the derivative of any of the

functions in the chain has the value zero, that is when any of functions has an extreme point or

has a horizontal tangent, then the composite function y = y{u(x)} will also have an extreme point

or have a horizontal tangent. We can compute the derivative of y = {sin(x)}2 to be

dy

dx=

dy

du∗

du

dx = 2 sin(x) cos(x) .

In figure 21 it is observed that whenever sin (x) has an extreme point (indicating a zero

derivative) then sin(x) 2 will also have an extreme point. In addition, whenever sin (x) has a

single root (zero crossing) then sin(x) 2 will have a double root, which is a minimum.

Figure 21 The composition of squaring the function, sin(x)

y = sin(x)

y = sin (x)2

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Differentials and infinitesimals

It may be appropriate here to clarify some prevailing mathematical misconceptions regarding the

notation of dx and dy for differentials. The differentials, dx and dy, are ordinary variables used

to represent displacements on a tangent line from the point of tangency. There is no restriction

on their size. The notation for the derivative, {dy

dx} represents a multiplier of the displacement

variable, dx, to obtain the associated displacement, dy, on the tangent line. {dy

dx} is not a

division. Just as in the equation y = mx + b there is no restriction on x; there is also no

restriction on dx in the equation dy ={dy

dx} dx. The variables, dx and dy can have the values of

100 million or 0, the equation still holds. As for infinitesimals, there appears to no need to

confuse beginning students with such concepts that are larger than zero but less than everything

else. In the system of real numbers, either a number is zero or it is not! If it is not zero, there are

other numbers between it and zero.

Single equation forms that describe curves

In the previous paragraph we looked at the composite or chain form that might describe the

relationship between the variables x and y. This form requires more than one equation. The

simple curves such as the polynomials and the rational curves can be described with one

equation where the vertical variable is isolated on the left in the form y = f(x). This form is called

explicit. However, for general algebraic curves it might not be possible to isolate the variables.

But then the curve can be described by a single equation which is a linear combination of

products of powers of x and y as in the equation:

xy7 – 2x

5y + x

3y

3 + 1 = 0.

This form where the variables, x and y, are knotted together is called implicit. If an equation

where the variables are related implicitly can be solved for x, then a form of the curve can be

derived, x = g(y), which is called the inverse form of the relationship between x and y. Note

that both the inverse and explicit forms are variations of the same implicit equation with the

same graph. If y is chosen as the independent variable then x = g(y) becomes the explicit form

and y = f(x) becomes the inverse form. Whichever of the two forms is selected to be explicit, the

other becomes the inverse form. In the next section we will develop the rule for computing the

slope of the tangent line to a curve which is described by an inverse form.

The derivative of a function described as an inverse form

Between a function minimum and the next maximum the function rises monotonically. And

between a function maximum and the next minimum the function descends monotonically. On

these intervals the function is said to be one-to-one meaning for each x there is only one y and

for each y there is only one x. In this case, the variable, y is a function of x ,say f(x) and in

addition the variable x is usually a different function of y, commonly denoted by x = f –1(y).

Since the – 1 superscript in the past has been used to mean reciprocal, it is unjustified and wrong

to now use this notation to mean the inverse function. With the trig functions the inverses are

ETD 315



called arc sine, etc. The two functions y(x) and x(y) should be called an inverse pair. Both the

explicit and inverse function forms are special cases of the implicit form.

If we start with x, compute y and then apply the inverse function to y, we will be returned to the

original value of x which would be represented in algebraic notation:

x = f –1{f(x)}. Equation 16

If we start with y and use the inverse function to compute x to which we then apply the function,

f, we will be returned to the original value of y which would be represented in algebraic notation:

y = f {f–1 (y)}. Equation 17

Examples of inverse pairs are:

1) y = 5x – 4 and x = .2(y + 4) , figures 22 and 23

2) y = x3 and x = y(1/3) figures 24 and 25

3) y = x2 and x = +√y or x = – √y figures 26 and 27 and 28

4) y = ln(x) and x = ey figures 29 and 30.

In the graphs for the inverse functions below, the variable, y, is plotted horizontally and the

variable, x, is plotted vertically but the explicit and inverse forms describe the same relationship

between the variables.

Figure 22 A straight line in explicit form Figure 23 The inverse of the straight line

x

y

y = 5x - 4

y

x

x =.2(y+4)The inverse function of y = 5x -4

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Figure 24_An odd degree power function Figure 25 The inverse of an odd degree

power function

Figure 26 A 2nd degree power function Figure 27 The square root function

Figure 28 The negative branch of the square root function

y = x + sin(5x))

The odd degree power function

y = x 3

x

y

x

y

y = x

y = x 2

x

y

y

x

x = the negative branch

The inverse function of y = x2

of the square root of y

ETD 315



Figure 29 y = ln(x) Figure 30 x = ey

The strategic importance of the inverse function is that to solve a condition where the unknown

value, say x, is buried in the argument of a function the inverse function will bring the unknown

into the open. If we want the value of x such that x2 = 100. The square root which is the inverse

to y = x2 will provide x = √100 = 10.

Examine a straight line which is described by the equation y = mx + b. The derivative, dy

dx , is

the slope m. The inverse function will be x = 1

m(y − b) whose derivative is

dx

dy =

1

m. For

straight lines the slope, dx

dy , at any point on the graph of the inverse function is the reciprocal of

the slope of the explicit form, dy

dx.

For a general curve we need only to apply the chain rule to either of the preceding two equations

to derive a formula for the derivative of the inverse form of a function. The notation, dy

dx , will be

used to represent the derivative of the function f(x) and the notation dx

dy will be used to represent

the derivative of the inverse function, f–1 (y). If we differentiate the function in Equation 16 with

respect to x and differentiate the function in Equation 17 with respect to y we find:

1 = dx

dx =

dx

dy dy

dx . and 1 =

dy

dy =

dy

dx dx

dy .

The derivatives, dy

dx and

dx

dy at any point on a curve are reciprocals and the notation suggests the

cancellation of differentials as would be done with ordinary fractions. Since a smooth curve has

only one tangent line at a point, the formula for the derivative of the inverse function is

y

x =( y+4 ) / 5

y = ln(x)

x

y

y = x - 1

1 2 3 4

y

x

x = e

The inverse function of y = ln(x)

2

y

ETD 315



dx

dy=

1

{ dy

dx } Equation 18

As an example, let us consider the pth root of the variable, y, which is the inverse function of the

power function, y = x p . The inverse form of the power function, y = x p is x = y1

𝑝 . The

derivative of the power function is dy

dx = p xp ─ 1. The derivative of the pth root is

dx

dy ={

dy

dx}─ 1 = ( p x p ─1) ─ 1 =

1

p x 1─ p =

1

p

x

xp =

1

p 𝑦

1

𝑝

y =

1

py

1

𝑝−1

.

We see that the derivative of the root function follows the same rule as the derivative of integer

powers. (Multiply by the exponent and decrease the exponent by 1.)

Differentiating fractional powers of x

We are now positioned to derive a rule for those algebraic functions which are fractional powers

of x,

y(x) = x q

p = (x 1

p)q,

where q and p are whole numbers. This case provides an excellent example of the application of

the chain rule. Invoking the chain rule we obtain

d

dx{x

q

p} =d

dx{x

1

p}q

= {q(x 1

p)(q−1) }{ 1

p x

1

p−1

}

= 1

p q x

(𝑞

𝑝−

1

𝑝+

1

𝑝−1)

= q

p x

(q

p – 1)

Equation 19

It is interesting to recognize that, here also, the derivative of the fractional power functions

follows the same rule as for differentiating the integer power functions. (Multiply by the

exponent and decrease the exponent by 1.) Note that the inverses of the odd power functions are

always single valued, but the branches of the even roots will have to be treated separately. See

figures 25, 27 and 28 showing the functions y = x1

3 , y = +x1

2 = + √x and y = − x1

2 = −√x .

Implicit Cartesian forms of algebraic curves and computing the slopes of their tangent lines

Algebraic curves in Cartesian coordinates can be described by a single equation whose implicit

form is a linear combination of products of powers of the two variables x and y. Examples are

the circle,

x 2+ y 2 – 100 = 0

and the tilted ellipse which is shown in Figure 31.

x 2 – xy + y 2 – 16 = 0. Equation 20

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Figure 31 A tilted ellipse

As was said above algebraic curves are generally multi-valued and cannot be described

explicitly. But they can be described in a single equation implicit form whose branches may be

described by explicit functions which at each point have only one tangent line. To find the

derivative of y with respect to x for a branch of an implicit algebraic form we will need to

consider {y(x)}n as a composite form and apply the chain rule as:

d

dx{y(x)}n = n {y(x)}(n-1)

dy

dx

We will compute the derivative of the upper branch of the tilted ellipse as an example. Set the

sum of the derivatives of each of the four terms below equal to zero. The product rule must be

applied to the xy term. In our example the implicit form is:

x2 – xy + y2 – 16 = 0

and then after differentiating term by term we obtain:

2x – y – xdy

dx + 2y

dy

dx= 0 .

The implicit derivative equation will always be linear in dy

dx which will enable us to solve for the

derivative, dy

dx :

dy

dx =

y – 2x

2y – x .

We should note that to evaluate the slope of the tangent line, both coordinates, x and y, must be

known. The ellipse will have a horizontal tangent when y = 2x and a vertical tangent when

y = 1

2 x as shown in figure 31.

More on forms

We have found procedures for computing the three single equation forms for functions. We have

viewed the chain rule for differentiating the composite form. There are other forms which are

conventionally treated in calculus courses: The Maclaurin, Taylor and Fourier infinite series

x

xx - xy + yy =16 y = 2x

y = x/2

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forms. There is the polar form where a special point called the origin is selected and a special

direction which is usually the positive horizontal axis is also selected. The implicit equations for

the polar form relate for each point on the curve, the distance to the origin, r, and the angle with

the horizontal, θ. Commonly θ is taken for the independent variable. To keep this paper brief, it

will remain to the student to research these topics in a conventional text when the need arises.

But one last important form, the parametric form, which is especially useful for the multi-

valued algebraic curves that loop or double back, should be considered here. Parametric forms

are not only well suited for describing multi-valued curves but extend naturally to curves in three

or higher dimensional spaces.

Parametric forms for curves in 2-space and the rule for their differentiation

The parametric form of a curve in the 2 dimensional Cartesian x-y space is another two equation

representation with an intermediate variable which is called the parameter. The equations

assign to each value of the parameter, a point, P(x, y) on the curve. If the parameter is called s,

the parametric equations would be

x = f(s) and y = g(s)

where f(s) and g(s) are single valued functions. Each value of the parameter will then refer to a

specific point on the curve, and varying the parameter will cause the point to move along the

curve.

If the parameter was time, then the equations could be considered as the position of a particle

moving along a trajectory. Any one-to-one function of the parameter can be used to provide a

second parameter occurring in a second different set of parametric equations. Of course, both

sets of parametric equations describe the same curve. And of course, both sets of parametric

equations yield the same tangent line at the same point of the curve.

If the range of the parametric coordinates does not span the entire curve, then the parametric

form will only describe a segment of the curve. As an example, say we choose for the parabola, y

= x2, the parametric form with the set of equations:

x = sin(t) and y = sin2(t)

Since as the variable, t, increases, the values of sin(t) will oscillate between plus and minus 1.

This will cause the point P(x, y) to move on the parabola oscillating around the vertex in the

interval –1 ≤ x ≤ 1. The motion of the point pauses and reverses direction at each of the

endpoints, x = ± 1.

To find the direction of the curve at any point, eliminate the parameter between the two

describing equations and consider y as a function of x. Then apply the chain rule to compute,

dy

dx =

dy

ds

ds

dx and then apply the inverse function rule,

ds

dx=

1

{ dxds

} , to obtain the rule for

computing the slopes of the two dimensional curve in the x – y space described by the parametric

form;

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dy

dx =

dy

dsdx

ds

. Equation 21

Derivatives of kinds of functions and directions of the corresponding curves

These derivative rules concerning the forms, operations and the power functions will permit us to

find the slopes of “not extremely” complicated curves of the following kinds 1) the curves of

polynomials, 2) the curves of rational functions and 3) algebraic curves. It remains to find rules

for the derivatives of 1) the trig functions and 2) the inverses of the trig functions and 3) the

exponential functions and 4) the log functions which are the inverses of the exponential

functions. The derivatives of the trig functions can be used to compute the directions of many

interesting algebraic and transcendental curves including the cycloids, the cardioids, the limaçon

and the astroids. A student taking calculus should immediately rush to see the Internet

animations of these curves, such as those provided by Wolfram. It is unfortunate that students

first see these curves as exercises in the middle of some text instead of being informed that the

17th and 18th century study of these curves provided a stimulus for the development of calculus.

The cycloid is obtained by following a point on a circle rolling on a straight line. The astroid is

obtained by following a point on a circle rolling inside a second larger circle. The cardiod is

obtained by following a point on a circle rolling outside a second circle of the same size. See

graphs of the cycloid, the astroid and the cardioid in figures 32, 33 and 34.

Figure 32 The cycloid

x

y

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Figure 33 The astroid Figure 34 The cardioid

The derivatives of the functions y = sin x, y = cos x and y = tan x

Start with the unit circle and consider the angle, θ in radians, that the radius drawn to a point P

on the circle makes with the horizontal axis. From geometry we know that the tangent to the

circle at P is perpendicular to this radius and so makes an angle of θ + π/2 radians as is shown in

figure 35. The slope of the radius is y

x and the perpendicular slope, which is the slope of the

tangent line will be – x

y . The differential change in distance on the tangent line caused by a

differential change in arc length, Δs, along the circle due to a differential change in angle, Δθ, is

Δs = Δθ radians. In figure 35 we can see the horizontal and vertical differential changes on the

tangent line are:

y2 – y1 = Δy = Δ sin θ = cos(θ) Δs = cos(θ) Δθ. Equation 22

x2 – x1 = Δx = Δ cosθ = – sin(θ) Δs = – sin(θ) Δθ. Equation 23

Figure 35 The relationship between the vertical and horizontal differentials and the circular arc.

x

y

x

y

A

A

P

P

rise = dy = d sin(A) = cos(A) * dA

run = dx = d cos(A) = -sin (A) * dA

2

1

dAd sin(A)

d cos(A)

The radius equals 1.

x = cos(A)

y = sin(A)

dA

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From equation 22 it is seen that on the tangent line the ratio of the vertical changes which are the

changes in sin(θ) due to the changes in θ is cos(θ). So,

d sin(θ)

dθ= cos (θ) Equation 24

Similarly from equation 23 it is seen d cos(θ)

dθ= −sin(θ) Equation 25

A second derivation of the formulas for the derivatives of the sine and cosine functions

Say the angle to the origin from a point P on a unit circle is θ measured in radians. Then

Δy = Δ{sin(θ) }= y2 – y1 = sin(θ + Δθ) – sin(θ ) = sin(θ)cos(Δθ) + sin(Δθ) cos(θ) – sin(θ)

and similarly

Δy = Δ{cos(θ) }= y2 – y1 = cos(θ + Δθ) – cos(θ ) = cos(θ)cos(Δθ) – sin(Δθ) sin(θ) – cos(θ) .

Figures 36 a and b The tangent lines of sin(x) and cos(x) at the origin

Referring to figure 36a, it is seen that at the origin y = sin(θ) is tangent to y = θ so both have the

same slope equal to 1. Also, it is seen in figure 36b that at the origin, y = cos(θ) is tangent to

both the line y = 1 and to the quadratic y = 1 – θ 2 / 2 all having a derivative of zero. Then

when we replace cos(Δθ) and sin(Δθ) by their tangential approximations we find

Δ{sin(θ)} = sin(θ)(1 – Δθ 2 / 2) + Δθ cos(θ) – sin(θ) = cos(θ) Δθ – sin(θ) Δθ 2 / 2) and

Δ{cos(θ)} = cos(θ) )(1 – Δθ 2 / 2) – Δθ sin(θ) – cos(θ) = – sin(θ) Δθ – cos(θ) (Δθ 2 / 2) .

x

y

Radians

y = x

y = sin(x)

x

y

y = cos(x)

y = 1

y = 1 - xx/2

ETD 315



The quadratic terms in Δθ which describe the deviations from tangency can be discarded and we

find

Δy = Δ{sin(θ) }= y2 – y1 = cos(θ ) Δθ and similarly, Δy = Δ{cos(θ) }= y2 – y1 = – sin(θ) Δθ)

The derivatives which are the multipliers of Δθ are seen to be:

d

dθsin θ = cos θ and

d

dθcos θ = sin θ,

confirming equations 24 and 25.

One last check can be made. Since the cosine is functionally related to the sine, the formula for

the derivative of the cosine can be deduced from the formula for the derivative of the sine. We

can apply the principal that the derivatives of identities are also identities to the trig identity

{sin(θ)} 2 + {cos(θ)} 2 = 1.

to get 2 sin(θ) d

dθsin θ + 2 cos(θ)

d

dθcos θ = 0

and then sin(θ) cos(θ) + cos(θ) d

dθcos θ = 0.

Cancelling the cosine factors leads to

d

dθcos θ = – sin(θ)

which serves to verify equation 25.

We have discovered the derivative of the sine is the cosine which has the same shape as the sine

but leads the sine by 90°. In addition, we find that the derivative of the cosine is the negative

sine which has the same shape as the cosine but leads the cosine by 90°. We know from

trigonometry that all sinusoids of the same frequency can be described as a linear combination of

a sine and a cosine of that same frequency, as in

C cos(2πf t + θ) = Acos(2πf t) + Bsin(2πf t). Equation 26

Applying the chain rule to the shifted sinusoid y(t) = C sin(2πf t + θ), leads to:

dy

dt = 2πf C cos(2πf t + θ)

which asserts that the derivative of any sinusoid is another sinusoid whose amplitude is increased

by 2πf and which is advanced in phase by 90° as shown in figure 37. Note that differentiating the

function y = sin(x) four times will advance the curve by 4 times 90° equaling 360° and return the

original sine curve.

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Figure 37 The sine curve and two derivatives

Since the function, tan(θ) = sin(θ)

cos (θ) is a quotient of the sin and cosine, the derivative of the

tangent function can be obtained by applying the quotient rule, Equation 11.

Finding the direction of the tangent line to a circle – an example solved in each of the forms

Let the center of a circle of radius 10 be located at the origin. Then the implicit form of the

circle is x2 + y

2 = 100. For each form we will compute the slope at the point P(8, 6). We will

begin by applying the principle from geometry that we have just used. If the radius drawn to the

point of tangency has a slope of y

x , then the tangent line will have a slope of –

x

y . If the point

on the circle is 8, 6, the slope of the radius will be 6

8=

3

4 and the slope of the tangent line will

be −4

3 as shown below in figure 38 .

We will first write the equation of the circle in the explicit form, y = √100 − x2 . This form

can be viewed as a composite: the square root of a parabola: y =√u ; u = 100 − x2 to which the

chain rule can be applied.

dy

dx=

dy

du du

dx=

1

2u

−1

2 (−2x) =− x

√u =

−x

√100−x2 = −

x

y Equation 27

Not surprisingly this value matches the value obtained by the geometric argument above.

x

y

y = sin(x)

y = -sin(x)

y = cos(x)

ETD 315



Figure 38 The tangent line to a circle

Considering the circle in parametric form and using as the parameter, the angle, θ, between the

radius of each point and the horizontal, the parametric equations for the circle become:

y = 10 sin θ and x = 10 cos θ .

We can check that the curve of this parametric form is a circle of radius 10 by eliminating the

parameter from the two equations. By substituting sin θ and cos θ into the trig identity

sin2 θ + cos

2 θ = 1, the parameter, θ, can be eliminated to obtain the implicit form for the

circle,

x2 + y

2 = 100.

The slope of the tangent line, dy

dx can be obtained in terms of the parameter, θ, by differentiating

both equations and eliminating the variable dθ.

dy = 10 cos θ dθ ; dx = – 10 sin θ dθ

dy

cos θ = 10 dθ = –

dx

sin θ=

dy

dx = –

cos θ

sin θ = –

x

y Equation 28

If our circle is described in inverse form with the equation x= √100 − y2 , then following the

procedure for the differentiating the explicit form we obtain dx

dy=

−y

√100−y2 =

−y

x . The

derivative can be found by taking the reciprocal of dx

dy which is

dy

dx= −

x

y. Equation 29

x

y

y = (3/4)x

P(8, 6)

x + y = 1002 2

(y - 6) = -(---)(x - 8)4

3

The radius equals 10

ETD 315



The easiest technique for evaluating the derivative is to work directly in the implicit form.

Differentiating the equation x2 + y

2 = 100 leads to 2x + 2y

dy

dx = 0 and then to

dy

dx= –

x

y. Equation 30

We can see from equations 27 to 30 that no matter in which form the circle is described, the

values of the slopes of the tangent lines agree with the value found by geometry, – x

y.

The derivative of the arcsin and arccos functions

Figure 39 y = arcsin(x) Figures 40 y = arccos(x)

The graphs of the arcsin and arccos functions are shown in figures 39 and 40. Both functions are

continuous and smooth in the interval, –1 < x < 1. While the arcsin(x) curve is seen to be

monotonically increasing, the arccos(x) curve monotonically decreases. Both curves are

observed to have infinite slopes at the ends of their domains. We need only apply the inverse

function rule to obtain the derivative of y = arc sin(x). Then x = sin(y) and

dx

dy = cos(y) and

dy

dx=

d

dxarc sin(x) =

1

cos (y) =

1

√1−𝑥2 Equation 31

Similarly if y = arc cos(x) then x = cos(y); dx

dy = −sin(y) and

dy

dx=

d

dxarc cos(x) =

−1

sin 𝑦 =

−1

√1−𝑥2 Equation 32

We should note that while the arcsine and arccosine functions are transcendental, their

derivatives are algebraic. The derivative of the arcsine is positive, that of the arccosine is

negative. The formulas, equations 31 and 32, for the derivatives of these functions confirm that

the slopes will be infinite when x = ±1.

y

x = ln(y)

y

y

x

y = arcsin(x)

1-1

y

x =( y+4 ) / 5

x = ln(y)

y

y

x = y -1

x

y = arccos(x)

1-1

ETD 315



The derivative of the function y = ex

In figure 41 we see the graphs of the curve family y = ax depicted as a increases from 1.2 to 3.

The curves are smooth, rising monotonically, from being asymptotic to the x-axis in the 2nd

quadrant, intersecting the y-axis at y = 1 and continuing to rise monotonically to infinity in the

first quadrant. The vertical values of each of these curves are positive.

Figure 41 The curve family y = ax for 1.2 ≤ a ≤ 3 Figure 42 The curve y = ex

It is seen that in the first quadrant of figure 41 for fixed horizontal values, x, the vertical values,

y, increase as a increases. Also for fixed x, the slopes increase as a increases. There is a special

curve in the family which has slope equal to 1 when the curve crosses the y-axis, that is, when

x = 0. This curve has a value for the base a approximately equal to 2.7182818284 which is a

special frequently occurring transcendental number called e. The equation of this special curve

is y = ex. The tangent line to this curve has a y-intercept of 1 and a slope of 1. In figure 42 we

see the graph of the curve y = ex and the tangent line at x = 0, whose equation is y = x + 1.

To find the slope of the tangent line at any other point P(x, y) on the curve, set

Δy = ex + Δ x – ex = ex (e Δ x – 1).

Examining the equation of the curve y = eu when u = 0, we see the tangent line has the equation

y = u + 1.and so for Δx having a value of 0, the tangent line will be y = Δx + 1. Then

Δy = ex (e Δ x – 1) = ex (Δx + 1 – 1) = ex Δx. It can now be seen that the slope of the tangent line,

which is the ratio of rise to run, on the tangent line, Δy

Δx , is ex .

d

dxex = ex Equation 33

Here we discover that the derivative of the function y = ex is also e

x. We should note here that

the chain rule predicts that the derivative of the function y = eax

will be dy

dx= ae

ax which has

the same shape as y = eax

but is stretched vertically by a factor of a.

x

y

y = ax

y = 1.2

y = 3.

x

x

x

y

y = ex

y = x + 1

y = 3.

ETD 315



The derivative of the natural log function y = ln(x)

Since the curve y = ex is continuous and smooth and monotonically rises from 0 to infinity, it has

an inverse which is called the natural log function, but which is not defined for negative values

of x. See figure 29. The natural log function has vertical values starting at − ∞ for small positive

values of x and which increases monotonically, continuously and smoothly toward + ∞ as x

increases.

We will now derive the rule for differentiating the natural log function. Since the function

y = ln(x) is the inverse of the function x = ey the rule for its derivative can be obtained by

applying the rule for inverse functions. Let x = ey . Then the inverse function is y = ln(x) and

the chain rule states dy

dx=

1

{ dx

dy }

and dx

dy = ey = x .

d

dxln (x) =

1

{ dx

dy } =

1

x . Equation 34

We discover from equation 34 that the slope of the function y = ln(x) varies in inverse proportion

to the horizontal coordinate that is that ln(x) is an increasing function but whose rate of increase

decreases toward 0 as the variable, x increases toward infinity.

Summary

The preceding pages provided a visual review of the basic rules of the differential calculus of

two dimensional curves. Functions are viewed as the single valued branches of well-behaved

curves. Inverse function – function pairs are one-to-one, forward and backward monotonic

segments of the curves. The derivative is defined simply as the slope of the tangent line at a point

on the curve, which is how Lagrange4 and the other 18th century developers of calculus viewed

the concept, and which is also the way most engineers use the concept today. All the derivative

rules conventionally treated in a conventional calculus course can be derived based on the idea

that the derivative is the slope of the tangent line, Δy = dy

dx Δx. There is no need at the beginning

of the study to define the derivative as a result of a limiting process employing deltas and

epsilons. This limiting process definition is a confusing roadblock to the analytical

understanding of too many students, which curtails STEM curriculum enrollments. The

mathematics teaching community must consider the value to the STEM community of delaying

the conventional delta-epsilon limiting processes until later in the curriculum.

Conclusion; Mathematical Proof and Pedagogy

In the realm of logical proof the community of mathematicians maintains the gold standard.

However the premature emphasis on proof in classrooms and in texts has been at the expense of

exposing a wider audience of prospective engineers and technicians to analytical concepts and

ETD 315



techniques. Only infrequently can engineering design be granted the certainty of mathematical

proof; more often engineers employ their judgment based on legal regulations and past

experience and make tradeoffs. Too many mathematical concepts are defined in ways that make

proving easy, not in the ways these concepts will be used subsequently. This paper serves as an

example that the rules of differential calculus can be introduced in the manner that engineers will

need in performing engineering calculations and without resorting to delta-epsilon arguments.

The teaching mathematicians have known that there are problems with math pedagogy, and over

the past half century or so have attempted various reform movements which have failed to

produce the numbers of analytically adept engineers our society needs. But the mathematicians

approach to reform has been cosmetic, maintaining the emphasis on proofs while hoping that

some application would keep students from bailing out. Applications should come after the

mathematics is understood and is best left to experts in their disciplines. Mathematics must serve

society as more than just a vehicle to display clever techniques of quantitative proof.

Ultimately it is up to the STEM community to form a consensus on how to introduce calculus to

K-12 teachers, so that more students in the next generation will feel comfortable with analytical

concepts and methods.

References

1. Aleksandrov, et al Mathematics Its Content, Methods and Meaning The MIT Press, Cambridge MA, 1963

2. Alsina, Claudi & Nelson, Roger B. Math Made Visual MAA, Washington, 2006

3. Arnold, Douglas N. http://www.ima.umn.edu/~arnold/calculus/secants/secants1/secants-g.html

4. Bressoud, David M. A Radical Approach to Real Analysis MAA, Washington, 2006

5. Dray, Tevian. Using Differentials to Differentiate Trigonometric and Exponential Functions

The College Mathematics Journal, Vol. 44 No. 1 January 2013 pg's. 18, 19

.6 Dray, Tevian. Putting Differentials Back into Calculus The College Mathematics Journal,

Vol. 41 No. 2 March 2013 pg's. 90 - 99

7. Grossfield, Andrew What are Differential Equations? A Review of Curve Families

ASEE Annual Conference, 1997

8. Grossfield, Andrew The Natural Structure of Algebra and Calculus ASEE Annual Conference, 2010

9. Grossfield, Andrew Mathematical Forms and Strategies ASEE Annual Conference, 1999

10. Grossfield, Andrew Mathematical Definitions: What is this thing? ASEE Annual Conference, 2000

11. Grossfield, Andrew Are Functions Real? ASEE Annual Conference, 2005

12. Grossfield, Andrew Visual Analysis and the Composition of Functions ASEE Annual Conference, 2009

13. Grossfield, Andrew Introducing Calculus to the High School Curriculum: Curves, Branches and Functions

ASEE Annual Conference, 2013

14. Grossfield, Andrew Visual Differential Calculus ASEE Zone 1 Conference, 2014

15. Nelson, Roger B. Proofs without Words: Exercises in Visual Thinking MAA Washington, 1993

16. Nelson, Roger B. Proofs without Words II: More Exercises in Visual Thinking MAA Washington, 2000

17. Dawkins, Paul http://tutorial.math.lamar.edu

18 University of Tennessee math Archives: http://archives.math.utk.edu/visual.calculus/

19. http://calc101.com/webMathematica/derivatives.jsp

20. Parris, Rick WINPLOT, a general-purpose plotting utility, Peanut Software

http://tutorial.math.lamar.edu/

http://archives.math.utk.edu/visual.calculus/

http://calc101.com/webMathematica/derivatives.jsp

ETD 315



ANDREW GROSSFIELD

Throughout his career Dr. Grossfield has combined an interest in engineering and mathematics. He earned a BEE at

the City College of New York and he obtained an M.S. degree in mathematics at the Courant Institute at night while

working full time as an engineer for aerospace/avionics companies. He studied continuum mechanics in the doctoral

program at the University of Arizona. He is a member of ASEE, IEEE and MAA.

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