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**2.
History of the Differential from the 17 ^{th} Century**

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The problem of finding the tangent to a curve has been studied by many mathematicians since Archimedes explored the question in Antiquity. The first attempt at determining the tangent to a curve that resembled the modern method of the Calculus came from Gilles Persone de Roberval during the 1630's and 1640's. At nearly the same time as Roberval was devising his method, Pierre de Fermat used the notion of maxima and the infinitesimal to find the tangent to a curve. Some credit Fermat with discovering the differential, but it was not until Leibniz and Newton rigorously defined their method of tangents that a generalized technique became accepted.

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**2.2
Roberval's Method of Tangent Lines using
Instantaneous Motion**

The primary idea behind Roberval's method of determining the tangent to a curve was the notion of Instantaneous Motion. That is, he considered a curve to be sketched by a moving point. If, at any point on a curve, the vectors making up the motion could be determined, then the tangent was simply the combination (sum) of those vectors.

Roberval applied this method to find the tangents to curves for which he was able to determine the constituent motion vectors at a point. For a parabola, Roberval was ableto determine such motion vectors.

Figure 2.1

Figure 2.1 depicts the graph of a parabola showing the constituent motion vectors V1 and V2 at a point P. Roberval determined that at a point P in a parabola, there are two vectors accounting for its instantaneous motion. The vector V1, which is in the same direction as the line joining the focus of the parabola (point S) and the point on the parabola (point P). The other vector making up the instantaneous motion (V2) is perpendicular to the y-axis (which is the directrix, or the line perpendicular to the line bisecting the parabola). The tangent to the graph at point P is simply the vector sum V = V1 + V2.

Using this methodology, Roberval was able to find the tangents to numerous other curves including the ellipse and cycloid. However, finding the vectors describing the instantaneous motion at a point proved difficult for a large number of curves. Roberval was never able to generalize this method, and therefore exists historically only as a precursor to the method of finding tangents using infinitesimals (Edwards 133-138).

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**2.3
Fermat's Maxima and Tangent**

Pierre De Fermat's method for finding a tangent was developed during the 1630's, and though never rigorously formulated, is almost exactly the method used by Newton and Leibniz. Lacking a formal concept of a limit, Fermat was unable to properly justify his work. However, by examining his techniques, it is obvious that he understood precisely the method used in differentiation today.

In order to understand Fermat's method, it is first necessary to consider his technique for finding maxima. Fermat's first documented problem in differentiation involved finding the maxima of an equation, and it is clearly this work that led to his technique for finding tangents.

The problem Fermat considered was dividing a line segment into two segments such that the product of the two new segments was a maximum.

Figure 2.2

In Figure 2.2, a line segment of length *a* is divided into two
segments. Those two segments are *x* and *(a - x)*. Fermat's
goal, then, was to maximize the product *x (a - x)*. His approach
was mysterious at the time, but with the benefit of the current knowledge
of limits, Fermat's method is quite simple
to understand. What Fermat did was to replace
each occurrence of *x *with *x + E* and stated that when the
maximum is found, *x *and *x + E *will be *equal*. Therefore,
he had the equation:

*x(a - x) = (x + E)(a - x - E)*

Through simplifying both sides of the equation and canceling like terms, Fermat reduced it:

At this point, Fermat said to simply let E = 0, and as such one is left with:

This says that to maximize the product of the two lengths, each length
should be half the total length of the line segment. Though this result
is correct, Fermat's method contains mysterious
holes that are only rectified by current knowledge. Fermat
simply lets E = 0, then in the step where he divides through by E, he would
have division by zero. However, though Fermat
formulated his method by saying E = 0, he was actually considering the
limit of E as it *approaches* zero (which explains why his algebra
works properly). Fermat's method of extrema
can be understood in modern terms as well. By substituting *x + E *for
*x*, he is saying that *f(x+E) = f(x)*, or that *f(x+E) - f(x)*
= 0. Since f(x) is a polynomial, this expression will be divisible by *E.
*Therefore, Fermat's method can be understood
as the definition of the derivative (when used for finding extrema):

Although Fermat was never able to make a logically consistent formulation, his work can be interpreted as the definition of the differential (Edwards 122-125).

Using his mysterious *E*, Fermat
went on to develop a method for finding tangents to curves. Consider the
graph of a parabola.

Figure 2.3

Fermat wishes to find a general formula
for the tangent to *f(x)*. In order to do so, he draws the tangent
line at a point *x* and will consider a point a distance *E*
away. As can be seen from Figure 2.3, by similar triangles, the following
relationship exists:

By isolating *s*, Fermat found
that

Fermat again lets the quantity *E
= 0 *(in modern term, he took the limit as *E* approached 0) and
recognized that the bottom portion of the equation was identical to his
differential in his method of mimina. Consequently, in order to find the
slope of a curve, all he needed to do was find f(x)/s. For example, consider
the equation :

Again, Fermat lets *E=0* and finds
that:

Now, returning to the original equation:

Here the modern notation for the derivative *f'(x) is *used, which
Fermat recognized to be equal to *[f(x+E)
- f(x)]/E *when he let *E=0.* Using this method, Fermat
was able to derive a general rule for the tangent to a function* *to
be . As described in the Integration
section, Fermat had now developed a general
rule for polynomial differentiation and integration. However, he never
managed to see the inverse relationship between the two operations, and
the logical inconsistencies in his justification left his work fairly unrecognized.
It was not until Newton and Leibniz
that this formulation became possible (Boyer 155-159).

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Newton and Leibniz served to complete three major necessities in the development of the Calculus. First, though differentiation and integration techniques had already been researched, they were the first to explain an "algorithmic process" for each operation. Second, despite the fact that differentiation and integration had already been discovered by Fermat, Newton and Leibniz recognized their usefulness as a general process. That is, those before Newton and Leibniz had considered solutions to area and tangent problems as specific solutions to particular problems. No one before them recognized the usefulness of the Calculus as a general mathematical tool. Third, though a recognition of differentiation and integration being inverse processes had occurred in earlier work, Newton and Leibniz were the first to explicitly pronounce and rigorously prove it (Dubbey 53-54).

Newton and Leibniz both approached the Calculus with different notations and different methodologies. The two men spent the latter part of their life in a dispute over who was responsible for inventing the Calculus and accusing each other of plagiarism. Though the names Newton and Leibniz are associated with the invention of the Calculus, it is clear that the fundamental development had already been forged by others. Though generalizing the techniques and explicitly showing the Fundamental Theorem of Calculus was no small feat, the mathematics involved in their methods are similar to those who came before them. Sufficiently similar are their methods that the specifics of their methodologies are beyond the scope of this paper. In terms of their mathematics, it is only their demonstration of the Fundamental Theorem of Calculus that will be discussed.

**2.5
The Ellusive Inverses – the Integral and Differential**

The notation of Leibniz most closely resembles that which is used in modern calculus and his approach to discovering the inverse relationship between the integral and differential will be examined. Though Newton independently arrived at the same conclusion, his path to discovery is slightly less accessible to the modern reader.

Leibniz defined the differential as being

From the earlier works of Cavalieri, Leibniz was already familiar with the techniques of finding the area underneath a curve. Leibniz discovered the inverse relationship between the area and derivative by utilizing his definition of the differential.

Consider the graph of the equation *y = x ^{2}+1*:

*Figure 2.4*

Leibniz’s idea was to use his differential on the area-function of the graph. Consider adding a D (area) underneath the graph of the curve. The D (area) is defined by the lower rectangle PQRS with area is y(D x) plus a fraction of the upper rectangle SRUT whose area is simply D x(D y). In other words, D (area) lies somewhere in between y(D x) and the total enclosing rectangle PQUT whose area is (y + D y )(D x). Leibniz then considered the ratio D (area)/ D x and saw that since the D (area) is between y(D x) and (y + D y)(D x) the ratio will be between y and (y + D y). From the diagram, it can be seen that D x and D y are closely related to each other. That is, as D x approaches 0 so too does D y. That means that the ratio D (area)/ D x lies between y and a value that approaches y (since y + D y approaches y as D y goes to 0). Written in terms of Leibniz’s definition of the derivative:

Leibniz has shown the inverse relationship
between the differential and the area-function. Namely that the differential
of the area-function of a function *y *is equal to the function itself.
In this case, the derivative of the area-function of *y = x ^{2}+1
*is indeed

Leibniz’s influence in the history
of the integral spreads beyond finding this groundbreaking relationship.
He was also responsible for inventing the notation that is used by most
students of calculus today. Leibniz used
the symbol ò (which was simply how "S"
was written at the time) to denote an infinite number of sums. This was
closely related to what he called the "integral", or the sum
of a number of infinitely small areas. The area underneath a function *y,
*or integral of *y,* was expressed as òy
(dx).

What Leibniz’s notation was really
saying was to sum up all of the areas dx * y as dx approached 0. As dx
approaches 0, there are an infinite number of such areas, hence the symbolism
ò representing an infinite number of
sums. Integration of this kind is also known as the *indefinite integral
*or *anti-derivative* due to the inverse relationship found by
Leibniz. That is, the derivative of the
indefinite integral of a function yields the function itself. . Leibniz
also developed a notation for *definite integrals*, or integrals which
produced the area underneath a curve between two bounding values (rather
than a symbolic answer). His notation for the *definite integral*
was to supply the lower and upper-bounding x-values with the integral symbol:

Where A is the area-function produced by the *anti-derivative*.
The area function A was computed by using Wallis’ law.

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