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Title: James Phillips's Lecture Notes, [18--]: Electronic Edition.
Author: Phillips, James, 1792-1867
Funding from the University Library, University of North Carolina at Chapel Hill supported the electronic publication of this title.
Text transcribed by Bari Helms
Images scanned by Bari Helms
Text encoded by Brian Dietz
First Edition, 2005
Size of electronic edition: ca. 10K
Publisher: The University Library, University of North Carolina at Chapel Hill
Chapel Hill, North Carolina
2005
© This work is the property of the University of North Carolina at Chapel Hill. It may be used freely by individuals for research, teaching and personal use as long as this statement of availability is included in the text
The electronic edition is a part of the University of North Carolina at Chapel Hill digital library, Documenting the American South.
Languages used in the text: English
Revision history:
2005-10-18, Brian Dietz finished TEI/XML encoding.
Source(s):
Title of collection: Cornelia Phillips Spencer Papers (#683), Southern Historical Collection, University of North Carolina at Chapel Hill
Title of document: James Phillips's Lecture Notes, [18--]
Author: [James Phillips]
Description: 1 page, 1 page image
Note: Call number 683 (Southern Historical Collection, University of North Carolina at Chapel Hill)
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James Phillips's Lecture Notes, [18--]
Phillips, James, 1792-1867



Page 1
ELEMENTS
of the
DIFFERENTIAL and INTEGRAL CALCULUS.

DIFFERENTIAL CALCULUS.
Of the differentiation of algebraic quantities.
1. We say that one variable is a function of another when the first is equal to a certain compound analytical expression of the second; for example, y is a function of x in the following equations:
y = (a2 - x2)½, y = x3- 3bx2, y = x4/a , y = l + cx2 , y = (a + bx + cx2 + dx3)m/n.
2. Let us consider a function in its state of augmentation, in consequence of the increase of the variable which it contains. As every function of a variable x can be represented by the ordinate of a curve BMM', let AP = x and PM = y be the ordinates of a point M of this curve, and let us suppose that the abscissa AP receives an increment PP' = h; the ordinate PM will become P'M' = y'. Fig. 1. To obtain the value of this new ordinate, we see that it is necessary to change x into x + h, in the equation of the curve, and the value which this equation will then determine for y will that of y'.
For example, if we had the equation y = mx2, we should obtain y' by changing x into x + h, and y into y', and we would have
y' = m(x + h)2 = mx2 + 2mxh + mh2
3. Let us now take the equation y = x3 . . . . . (1),
and let us suppose that y becomes y' when x becomes x + h; we shall have
y' = (x + h)3 = x3 + 3x2h + 3xh2 + h3:
if from this equation we subtract equation (1), there will remain
y1 - y = 3x2h + 3xh2 + h3; and dividing by h
[y'/h - y/h] = 3x2 + 3xh + h2 . . . . . (2).
Let us see what this result teaches us: y' - y represents the increment of the function y in consequence of the increment h given to x, since this difference y' - y is that of the new state of magnitude of y, as respects its primitive state.
On the other hand the increment of x being h, it follows that [y'/h - y/h] is the ratio of the increment [...]