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ELEMENTARY TEXTBOOK
ON THE
CALCULUS
BY VIRGIL SNYDER, Ph.D.
AND
JOHN IRWIN HUTCHINSON, Ph.D Of Cornell University
NEW YORK •:• CINCINNATI •:• CHICAGO
AMERICAN BOOK COMPANY
THE MODERN MATHEMATICAL SERIES. Lucien Augustus Wait,
{Senior Professor of Mathematics in Cornell University ,)
General Editor.
This series includes the following works : BRIEF ANALYTIC GEOMETRY. By J. H. Tanner and Joseph Allen. ELEMENTARY ANALYTIC GEOMETRY. By J. H. Tanner and Joseph
Allen. DIFFERENTIAL CALCULUS. By James McMahon and Virgil Snyder. INTEGRAL CALCULUS. By D. A. Murray. DIFFERENTIAL AND INTEGRAL CALCULUS. By Virgil Snyder and
J. I. Hutchinson. ELEMENTARY TEXTBOOK ON THE CALCULUS. By Virgil Snyder
and J. I. Hutchinson.
HIGH SCHOOL ALGEBRA. By J. H. Tanner. ELEMENTARY ALGEBRA. By J. H. Tanner. ELEMENTARY GEOMETRY. By James McMahon.
COPYRIGttl', 1912, BY
AMERICAN BOOK COMPANY
EL. CALCULUS. W. P. 1
Q fV3 03
53/
PREFACE
The present volume is the outgrowth of the requirements for students in engineering and science in Cornell University, for whom a somewhat brief but adequate introduction to the Calculus is prescribed.
The guiding principle in the selection and presentation of the topics in the following pages has been the ever increasing pressure on the presentday curriculum, especially in applied science, to limit the study of mathematics to a minimum of time and to the topics that are deemed of most immediate use to the professional course for which it is preparatory.
To what extent it is wise and justifiable to yield to this pressure it is not our purpose to discuss. But the constantly accumulating details in every pure and applied science makes this attitude a very natural one towards mathematics, as well as towards several other subjects which are subsidiary to the main object of the given course.
This desire to curtail mathematical training is strikingly evidenced by the numerous recent books treating of Calculus for engineers, for chemists, or for various other professional students. Such books have no doubt served a useful purpose in various ways. But we are of the opinion that, in spite of the unquestioned advantages of learning a new method by means of its application to a specific field, a student would ordinarily acquire too vague and inaccurate a command of the fundamental ideas of the Calculus by this onesided presenta tion. While a suitable illustration may clear up the difficulties
3
262792
4/( • ;/f ;'l :*•/;• \ * ., 'breface
of an abstract theory, too constant a dwelling among applica tions alone, especially from one point of view, is quite as likely to prevent the learner from grasping the real significance of a vital principle.
In recognition of the demand just referred to, we have made special effort to present the Calculus in as simple and direct a form as possible consistent with accuracy and thoroughness. Among the different features of our treatment, we may single out the following for notice.
The derivative is presented rigorously as a limit. This does not seem to be a difficult idea for the student to grasp, espe cially when introduced by its geometrical interpretation as the slope of the line tangent to the graph of the given func tion. For the student has already become familiar with this notion in Analytic Geometry, and will easily see that the analytic method is virtually equivalent to a particular case of the process of differentiation employed in the Calculus.
In order to stimulate the student's interest, easy applications of the Differential Calculus to maxima and minima, tangents and normals, inflexions, asymptotes, and curve tracing have been introduced as soon as the formal processes of differentia tion have been developed. These are followed by a discussion of functions of two or more independent variables, before the more difficult subject of infinite series is introduced.
In the chapter on expansion, no previous knowledge of series is assumed, but conditions for convergence are discussed, and the criteria for determining the interval of convergence of those series that are usually met with in practice are derived.
A chapter on the evaluation of indeterminate forms and three chapters on geometric applications furnish ample illus
PREFACE 5
tration of the uses of infinite series in a wide range of problems.
By reason of its significance in applications, it does not seem advisable to omit the important principle of rates. Arising out of the familiar notion of velocity, it affords an early glimpse into applications of the Calculus to Mechanics and Physics. We do not propose to make the Calculus a treatise on Mechanics, as seems to be the tendency with some writers; but a final chapter on applications to such topics of Mechanics as are easy to comprehend at this stage is thought advisable and sufficient. Especially in treating of center of gravity, the formulas have been derived in detail, first for n particles, and then, by a limit ing process, for a continuous mass. This was considered the more desirable, as textbooks in applied mathematics frequently lack in rigor in discussing the transition from discrete particles to a continuous mass. Besides, the derivation of these formu las affords a very good application of the idea of the definite integral as the limit of a sum. This idea has been freely and consistently used in the derivation of all applied formulas in the Integral Calculus. However, as the formula for the length of arc in polar coordinates is especially difficult of derivation by this method, we have deduced it from the corresponding formula for rectangular coordinates by a transformation of the variable of integration.
Inorder to make the number of new ideas as few as possible, the notions of infinitesimals and orders of infinitesimals have been postponed to the last article on Duhamel's principle. This principle seems to flow naturally and easily from the need of completing the proof of the formulas for center of gravity. The teacher may omit this article, but its presence should at
6 PREFACE
least serve the important end of calling the attention of the student to the fact that there is something yet to be done in order to make the derivations complete.
Some teachers will undoubtedly prefer to do a minimum amount of work in formal integration and use integral tables in the chapters on the applications. For such the first chapter of the Integral Calculus might suffice for drill in pure integration. The problems in this chapter are numerous, and, for the most part, quite easy, and should furnish the student a ready insight into the essential principles of integration.
The characteristic features of the books on the Calculus previously published in this series have been retained. The extensive use of these books by others, and a searching yearly test in our own classroom experience convince us that any far reaching change could not be undertaken without endangering the merits of the book. The changes that have been made are either in the nature of a slight rearrangement, or of the addi tion of new illustrative material, particularly in the applications.
We wish to acknowledge our indebtedness to our colleagues, who have added many helpful suggestions ; to Professor I. P. Church, of the College of Civil Engineering, for a number of very useful problems in applications of integration (See Exs. 1418, pp. 318320, and Exs. 67, pp. 323324), and particu larly to Professor James McMahon, who has carefully read all the manuscript, assisted throughout in the proof reading, and made many improvements in the text.
CONTENTS
DIFFERENTIAL CALCULUS CHAPTER I
Fundamental Principles
ARTICLE PAGE
1. Elementary definitions 15
2. Illustration : Slope of a tangent to a curve . . . .16
3. Fundamental theorems concerning limits 17
4. Continuity of functions 19
5. Comparison of simultaneous increments of two related variables 20
6. Definition of a derivative 21
7. Process of differentiation 22
8. Differentiation of a function of a function .... 23
CHAPTER II Differentiation of the Elementary Forms
9. 
Differentiation of the product of a constant and a variable 
25 
10. 
Differentiation of a sum . 
26 
11. 
Differentiation of a product 
27 
12. 
Differentiation of a quotient 
28 
13. 
Differentiation of a commensurable power of a function . 
29 
14. 
Differentiation of implicit functions 
33 
15. 
Elementary transcendental functions .... 
34 
16. 
Differentiation of loga x and loga u 
34 
17. 
Differentiation of the simple exponential function 
36 
18. 
Differentiation of an incommensurable power . 
37 
19. 
Limit of ^— as 6 approaches 0 
38 
20. 
Differentiation of sin u 
39 
21. 
Differentiation of cos u 
40 
22. 
Differentiation of tan u 
40 
23. 
Differentiation of sin1 u 
42 
24. 
Table of fundamental forms 
44 
CONTENTS
CHAPTER III
Successive Differentiation
ARTICLE PAGE
25. Definition of the nth derivative 47
26. Expression for the nth derivative in certain cases ... 49
CHAPTER IV Maxima and Minima
27. Increasing and decreasing functions 51
28. Test for determining intervals of increasing and decreasing . 51
29. Turning values of a function 53
30. Critical values of the variable 55
31. Method of determining whether <f>'(x) changes its sign in pass
ing through zero or infinity ....... 55
32. Second method of determining whether <p'(x) changes its sign
in passing through zero ... ..... 57
33. The maxima and minima of any continuous function occur
alternately .59
34. Simplifications that do not alter critical values .... 59
35. Geometric problems in maxima and minima .... 60
CHAPTER V Rates and Differentials
36. Rates. Time as independent variable 68
37. Abbreviated notation for rates 72
38. Differentials often substituted for rates 74
39. Theorem of mean value 74
CHAPTER VI
Differential of an Area, Arc, Volume, and Surface of Revolution
40. Differential of an area 78
41. Differential of an arc 79
42. Trigonometric meaning of — , — .80
dx dy
43. Differential of the volume of a surface of revolution ... 81
CONTENTS
ARTICLE
44. Differential of a surface of revolution
45. Differential of arc in polar coordinates
46. Differential of area in polar coordinates
81 82 83
CHAPTER VII Applications to Curve Tracing
47. Equation of tangent and normal ....
48. Length of tangent, normal, subtangent, and subnormal
49. Concavity upward and downward 60. Algebraic test for positive and negative bending
51. Concavity and convexity toward the axis .
52. Hyperbolic and parabolic branches .
53. Definition of a rectilinear asymptote .
85 85 89 90 94 95 96
Determination of Asymptotes
54. Method of limiting intercepts 96
55. Method of inspection. Infinite ordinates, asymptotes parallel
to axes 97
56. Method of substitution. Oblique asymptotes .... 99
57. Number of asymptotes 102
Polar Coordinates
58. Meaning of p — 104
dp
59. Relation between ^ and p— 105
dx dp
60. Length of tangent, normal, polar subtangent, and polar sub
normal 105
CHAPTER VIII
Differentiation of Function
61. Definition of continuity
62. Partial differentiation
63. Total differential
64. Total derivative . •
65. Differentiation of implicit functions
66. Geometric interpretation .
67. Successive partial differentiation
68. Order of differentiation indifferent
s of Two Variables
109 110 112 115 116 118 122 122
10 CONTENTS
CHAPTER IX
Change of Variable
article page
69. Interchange of dependent and independent variables . . 124
70. Change of the dependent variable 125
71. Change of the independent variable 126
72. Simultaneous changes of dependent and of independent variables 126
CHAPTER X Expansion of Functions
73. Convergence and divergence of series 132
74. General test for convergence , 133
75. Interval of convergence 138
70. Remainder after n terms , . 140
77. Maclaurin's expansion of a function in a power series . .141
78. Taylor's series 148
79. Rolle's theorem 150
80. Form of remainder in Maclaurin's series 150
81. Another expression for the remainder 153
CHAPTER XI
t
Indeterminate Forms
82. Definition of an indeterminate form » 157
83. Indeterminate forms may have determinate values . . . 158
84. Evaluation by development 160
85. Evaluation by differentiation ....... 161
86. Evaluation of the indeterminate form g 165
CHAPTER XII Contact and Curvature
87. Order of contact 167
88. Number of conditions implied by contact 168
89. Contact of odd and of even order 169
90. Circle of curvature 172
91 . Length of radius of curvature ; coordinates of center of curvature 1 72
92. Limiting intersection of normals 174
93. Direction of radius of curvature . . . ' • • 175
CONTENTS
11
ARTICLE PAGE
94. Total curvature of a given arc ; average curvature . . . 176
95. Measure of curvature at a given point 177
. 96. Curvature of an arc of a circle 178
97. Curvature of osculating circle 178
98. Direct derivation of the expressions for k and B in polar co
ordinates 180
EvOLUTES AND INVOLUTES
99. Definition of an evolute 182
100. Properties of the evolute 184
CHAPTER XIII Singular Points
101. Definition of a singular point
102. Determination of singular points of algebraic curves
103. Multiple points
104. Cusps .........
105. Conjugate points
191 191 193 194 197
CHAPTER XIV Envelopes
106. Family of curves
107. Envelope of a family of curves
108. The envelope touches every curve of the family
109. Envelope of normals of a given curve
110. Two parameters, one equation of condition
200 201 202 203 204
12 CONTENTS
INTEGRAL CALCULUS CHAPTER I
General Principled oe Integration
ARTICLE PAGE
111. The fundamental problem 209
112. Integration by inspection 210
113. The fundamental formulas of integration .... 211
114. Certain general principles 212
115. Integration by parts 216
116. Integration by substitution 219
117. Additional standard forms 222
118. Integrals of the forms C(^x + B)dx and C (Ax + B)dx # »M
J ax2 + bx + c J ^/ax2 + bx + c
119. Integrals of the forms f — and
J (Ax + B) Vax2 + bx + c
dx
5
(Ax + B)2 Vax2 + bx + c
121. Decomposition of rational fractions
122. Case I. Factors of the first degree, none repeated
123. Case II. Factors of the first degree, some repeated
124. Case II J. Occurrence of quadratic factors, none repeated
125. Case IV. Occurrence of quadratic factors, some repeated
126. General theorem
226
CHAPTER II 120. Reduction Formulas 229
CHAPTER III
Integration of Rational Fractions
238 240 241 243 245 247
CHAPTER IV
Integration by Rationalization
127. Integration of functions containing the irrationality y/ax + b 248
128. Integration of expressions containing Vax2 + bx + c . . 249
129. The substitution V± t1 ± k2 z 253
CONTENTS
13
CHAPTER V
Integration of Trigonometric Functions
ARTICLE
130. Integration by substitution
181. Integration of ( sec2na; dx, \ csc2»x dx
132. Integration of I secTOx tan'2n+1x dx, j cscmx coV^+h; dx
133. Integration of i tan"x dx, \ cotnx dx
134. Integration of ( sin'"x cosnx dx
135. Integration of f — , f
J a + b cos nx J a
k
dx b sin nx
dx
+ b sin nx f c cos nx 136. Integration of I eax sin wxcJx, je^cos nxdx
PAGE
255 255
257
258
260
264 266
CHAPTER VI
Integration as a Summation. Areas
137. Areas 268
138. Expression of area as a definite integral 270
139. Generalization of the area formula 273
140. Certain properties of definite integrals 274
141. Maclaurin's formula 276
142. Remarks on the area formula  277
143. Precautions to be observed in evaluating definite integrals . 283
144. Calculation of area when x and y are expressible in terms of a
third variable 289
145. Areas in polar coordinates 291
146. Approximate integration. The trapezoidal rule . . . 292
147. Simpson's rule 294
148. The limit of error in approximate integration .... 295
14
CONTENTS
CHAPTER VII
Geometrical Applications
ARTICLE PAGE
149. Volumes by single integration 298
150. Volume of solid of revolution 302
151. Lengths of curves. Rectangular coordinates .... 306
152. Lengths of curves. Polar coordinates 309
153. Measurement of arcs by the aid of parametric representation . 310
154. Area of surface of revolution 312
155. Various geometrical problems leading to integration . 315
CHAPTER VIII Successive Integration
156. Functions of a single variable .
157. Integration of functions of several variables
158. Integration of a total differential
159. Multiple integrals . . .
160. Definite multiple integrals
161. Plane areas by double integration .
162. Volumes
321 324 326 328 329 330 334
CHAPTER IX
Some Applications of Integral Calculus to Problems of Mechanics
163. 
Liquid pressure on a plane vertical wall . 
. . . 338 

164. 
Center of gravity 
. 340 

165. 
Moment of inertia . 
. 346 

166. 
Duhamel's theorem . 
. 348 

Trigonometric Formulas 
. 352 

Logarithmic Formulas 
. 353 
DIFFERENTIAL CALCULUS
^XKc
CHAPTER I
FUNDAMENTAL PRINCIPLES
1. Elementary definitions. A constant number is one that retains the same value throughout an investigation in which it occurs. A variable number is one that changes from one value to another during an investigation. If the variation of a number can be assigned at will, the variable is called independent; if the value of one number is determined when that of another is known, the former is called a dependent variable. The depend ent variable is called also a function of the independent variable.
E.g., 3 x2, 4vx — 1, cos x, are all functions of x.
Functions of one variable x will be denoted by the symbols /(#), <f>(x), • ••, which are read as "/of x" " <f> of x" etc. ; simi larly, functions of two variables, x, y, will be denoted by such expressions as
f(?,y),F(x,y), ••••
When a variable approaches a constant in such a way that the difference between the variable and the constant may be come and remain smaller than any fixed number, previously assigned, the constant is called the limit of the variable.
15
16
DIFFERENTIAL CALCULUS
2. Illustration : Slope of a tangent to a curve. To obtain the slope of the tangent to a curve at a point P upon it, first take the slope of the line joining P = (xly yx) to another point (x2, y2) upon the curve, then determine the limiting value of the slope
m
as the second point approaches to coincidence with the first, always remaining on the curve.
Ex. 1. Determine the slope of the tangent to the curve
2FM + fc
at the point (2, 4) upon it.
Here, x\ = 2, y\ = 4. Let x2 = 2 + h, yi = 4 + k, where h, k are so related that the point (x2, y*) lies on the curve.
Thus 4 + k = (2 + h)\
or h = 4 A + A2 (1)
The 
slope 
m = y*  x2  
Xi 
becomes 
4+ Tc 2 + h 
4 _ 2 
k 
Fig. 1
which from (1) may be written in the form
k
= 4 + h.
(2)
The ratio k : h measures the slope of the line joining (xh yx) to (ar2, ys) • When the second point approaches the first as a limiting position, the first member of equation (2) assumes the indeterminate form , but the second member approaches the definite limit 4. When
the two points approach coincidence, a definite slope 4 is obtained, which is that of the tangent to the curve y = x2 at the point (2, 4).
It may happen that h, k appear in both members of the equation which defines the slope, as in the next example.
FUNDAMENTAL PRINCIPLES
17
Fig. 2
Ex. 2. If x2 + y1 — «2? find the slope of the tangent at the point Oi> yd Since
Xl* + i/!2 = a2, (asi + ny+ (t/1 + ky = a\
hence 2 hx1 + A2 + 2 /fr/i + fc2 = 0,
from which  = —  — — — h 2 ?/i + k
k To obtain the limit of , put h, k h
each equal to zevo in tlie second member.
lim * = _*!.
h±o h ?/i
This step is more fully justified in the next article. This result agrees with that obtained by elementary geome try. The slope of the radius to the circle a2 + y2 = a2 through
the point (xlf yx) is — , and the slope of the tangent is the nega
tive reciprocal of that of the radius to the point of tangency, since the two lines are perpendicular.
3. Fundamental theorems concerning limits. The following theorems are useful in the processes of the Calculus.
Theorem 1. If a variable a approaches zero as a limit, then lea will also approach zero, k being any finite constant.
That is, if a = 0,
then Jca = 0.
For, let c be any assigned number. By hypothesis, a can be come less than , hence ka can become less than c, the arbi
k ■ '
* For convenience, the symbol = will be used to indicate that a variable approaches a constant as a limit; thus the symbolic form x = a is to be read " the variable x approaches the constant a as a limit." el. calc — 2
18 DIFFERENTIAL CALCULUS
trary, assigned number, hence ka approaches zero as a limit. (Definition of a limit.)
Theorem 2. Given any finite number n of variables a, (3, y, •••, each of which approaches zero as a limit, then their sum will approach zero as a limit. For the sum of the n variables does not at any stage numerically exceed n times the largest of them, which by Theorem 1 approaches zero.
Theorem 3. If each of two variables approaches zero as a limit, their product will approach zero as a limit. More gen erally, if one variable approaches zero as a limit, then its product with any other variable having a finite limit will have the limit zero, by Theorem 1.
Theorem 4. If the sum of a finite number of variables is variable, then the limit of their sum is equal to the sum of their limits ; i.e.,
lim (x + y + • • •) = lim x + lim y + For, if x = a, y = b, • • •,
then x = a + a, y = b \ (3, ••,
wherein a = 0, fi = 0, •• • ; (Def . of limit)
hence x + y+ ••• = (o+ &+ •••) + (« + fi+ •••)>
but a + p+> =0, (Th. 2)
hence, from the definition of a limit,
lim (x + y + •••) = a\b\ ••• = lim x f lim y + •••.
Theorem 5. If the product of a finite number of variables is .variable, then the limit of their product is equal to the product of their limits.
For, let x = a + a, y = b+($,
wherein a = 0, (3 = 0,
so that lim x = a, lim y = b.
FUNDAMENTAL PRINCIPLES 19
Form the product
xy = (a + a)(b + fi) = ab + «6 f /3a + «0. Then lim sc?/ = lim (ab + ab + (5a + a(5)
= ab + lim ab + lim 0a + lim a/3 (Th. 2)
= ab. (Th. 1)
Hence lim xy = lim a; • lim y.
In the case of a product of three variables x, y, z, we have lim xyz = lim xy • lim z (Th. 5)
= lim x lim y lim 3, and so on, for any finite number of variables.
Theorem 6. If the quotient of two variables as, y is vari able, then the limit of their quotient is equal to the quotient of their limits, provided these limits are not both infinite or not both zero.
(Th. 5)
y lim y
4. Continuity of functions. When an independent variable x, in passing from a to b, passes through every intermediate value, it is called a continuous variable. A function f(x) of an independent variable x is said to be continuous at any value xl} or in the vicinity of xXi when f(x^) is real, finite, and determi nate, and such that in whatever way x approaches a^,
From the definition of a limit it follows that corresponding to a small increment of the variable, the increment of the
For, since 
X x = y, y 
lim x = lim  lim yy y 

and hence 
,. x lim x lim  = 
20
DIFFERENTIAL CALCULUS
function is also small, and that corresponding to any number c, previously assigned, another number 8 can be determined, such that when h remains numerically less than 8, the differ
is numerically less than c.
2/,+e
a>.+5
Fig. 3
Thus, the function of Fig. 3 is continuous between the values xx and xY f 8, while the functions of Fig. 4 and Fig. 5 are dis continuous. In the former of these two the function becomes infinite at x = c, while in the latter the difference between the value of the function at c + h and c — h does not approach zero with h, but approaches the finite value AB as h ap proaches zero.
When a function is continuous for every value of x between a and b, it is said to be continuous within the interval from a to b.
5. Comparison of simultaneous increments of two related vari ables. The illustrations of Art. 2 suggest the following general procedure for comparing the changes of two related variables. Starting from any fixed pair of values x1} y^ represented graph ically by the abscissa and ordinate of a chosen point P on a given curve whose equation is given, we change the values of
FUNDAMENTAL PRINCIPLES
21
x and y by the addition of small amounts h and k respectively, so chosen that the new values xL + h and yl + k shall be the coordinates of a point P2 on the curve. The amount h added to xx is called the increment of x and is entirely arbitrary. Likewise, k is called the increment of y ; it is not arbitrary but depends upon the value of h ; its value can be calcu lated when the equation of the curve is given, as is shown by equation (1). These increments are not necessarily positive. In the case of continuous functions, h may always be taken positive. The sign of k will then depend upon the function under consideration. The slope of the line
PjP2 is then  and the slope of the tangent line at Pj is the
limit of  as h and consequently k approach zero.
The determination of the limit of the ratio of k to h as h and k approach zero is the fundamental problem of the Differential Calculus. The process is systematized in the following ar ticles. While the related variables are here represented by ordinate and abscissa of a curve, they may be any two related magnitudes, such as space and time, or volume and pressure of a gas, etc.
6. Definition of a derivative. If to a variable a small incre ment is given, and if the corresponding increment of a contin uous function of the variable is determined, then the limit of the ratio of the increment of the function to the increment of the variable, when the latter increment approaches the limit zero, is called the derivative of the function as to the variable.
22 DIFFERENTIAL CALCULUS
k That is, the derivative is the limit of  as h approaches zero,
or
liin (k
For the purpose of obtaining a derivative in a given case it is convenient to express the process in terms of the following steps:
1. Give a small increment to the variable.
2. Compute the resulting increment of the function.
3. Divide the increment of the function by the increment of the variable.
4. Obtain the limit of this quotient as the increment of the variable approaches zero.
7. Process of differentiation. In the preceding illustrations, the fixed values of x and of y have been written with sub scripts to show that only the increments h, k vary during the algebraic process of finding the derivative, also to emphasize the fact that the limit of the ratio of the simultaneous incre ments h, k depends upon the particular values which the variables x, y have, when they are supposed to take these in crements. With this understanding the subscripts will hence forth be omitted. Moreover, the increments h, k will, for greater distinctness, be denoted by the symbols Ax, Ay, read " increment of x," " increment of y."
If the four steps of Art. 6 are applied to the function y = <£ (x), the results become y + £fy=<f>(x + \x),
Ay = <j>(x + Ax) — <f>(x) = A<f> (x),
Ay _<f>(x + Ax) — <j> (x) _ A<£ (x)
Ax Ax Ax
. lira AJ = Km {+(* + **) »(*)  =Um Aj>^
A# Ax Ax
' FUNDAMENTAL PRINCIPLES 23
The operation here indicated is for brevity denoted by the
symbol — , and the resulting derivative function by <f>'(x); thus dx
dy _d<f>(x) _ lim f <f>(x + Ax)<j>(x)'
dx dx Aa; = 0
Ax
= +(*).
The new symbol ^ is not (at the present stage at least) to ax
be looked upon as a quotient of two numbers dy, dx, but rather as a single symbol used for the sake of brevity in place of the expression " derivative of y with regard to x."
The process of performing this indicated operation is called the differentiation of <f> (x) with regard to x.
EXERCISES Find the derivatives of the following functions with regard to x.
5. I.
X3
6. xn, n being a positive integer.
72
7.
1. 
x2 2x, 2x\ 3; x. 
2. 
3x*4:x + 3. 
3. 
1 4*' 
4. 
**2 + i. X2 
9. y = Vx. 

10. y = x~$. 
8.
ar+1 x
f 1
[Put #2 = x, and apply the rules.]
8. Differentiation of a function of a function. Suppose that y, instead of being given directly as a function of x, is expressed as a function of another variable u, which is itself expressed as a function of x. Let it be required to find the derivative of y with regard to the independent variable x.
Let y =f(u), in which u is a function of x. When x changes to the value au + Aaj, let u and y, under the given relations,
24 DIFFERENTIAL CALCULUS
change to the values u + Aw, y + A?/. Then
A?/ _ Aw Aw ,
Aa; — Aw Ax
hence, equating limits (Th. 5, Art. 3),
dy _dy da _ df(u) du dx ~~ du dx~ du dx
This result may be stated as follows :
The derivative of a function of u with regard to x is equal to the product of the derivative of the function with regard to w, and the derivative of u with regard to x.
EXERCISES
1. Given v = 3u2l, M = 3x2 + 4; find ^
dy du
du dx
dx du dx K '
2. Given ?/ =3m24u+ 5,« = 2x35; find ^ •
3. Given y = ,w = 5a;22x + 4; find ^ •
1 r3 3 _ , rfv
3 m2 3 a:3 aar
CHAPTER II
DIFFERENTIATION OF THE ELEMENTARY FORMS
dv In recent articles, the meaning of the symbol f was ex
ctx
plained and illustrated ; and a method of expressing its value, as a function of x, was exemplified, in cases in which y was a simple algebraic function of x, by direct use of the definition. This method is not always the most convenient one in the dif ferentiation of more complicated functions.
The present chapter will be devoted to the establishment of some general rules of differentiation which will, in many cases, save the trouble of going back to the definition.
The next five articles treat of the differentiation of algebraic functions and of algebraic combinations of other differentiable functions.
9. Differentiation of the product of a constant and a variable.
Let 
y = cu, 
Then 
y + Ay = c(u + Au), 
A?/ = c(m + Au) — cu = cAu, 

Ay Au Ax Ax' 

therefore 
dy du dx~~ dx 
Thus 
d(cu) _ du dx dx 
25 
(1^
26 DIFFERENTIAL CALCULUS
The derivative of the product of a coyistant and a variable is equal to the constant multiplied by the derivative of the variable.
10. Differentiation of a sum.
Let 2/==M_[_<y_W;_j_ ...
in which u. v, w, ••> are functions of x.
Then y + Ay = u + Au + v + Av — w — Aw + • • •,
Ay = Au + A?; — Aiv + • • •,
Ay _ Au ,Av_ Aw
Ax Ax Ax Ax '
dy _du dv dw dx dx dx dx
Hence f(u + v w+ . ..)=f^+^^+ .. (2)
doc doc doc doc
The derivative of the sum of a finite number of fractions is equal to the sum of their derivatives.
Cor. If y = u + c, c being a constant, then y + Ay = u + Au + c, hence Ay = Au,
and dy = du
dx dx
This last equation asserts that all functions which differ from each other only by an additive constant have the same derivative.
Geometrically, the addition of a constant has the effect of moving the curve y = u(x) parallel to the yaxis ; this opera tion will obviously not change the slope at points that have
the same x.
c /rtN dy du , dc
From (2), f = — + — ;
dx dx dx
DIFFERENTIATION OF THE ELEMENTARY FORMS 27
but from the fourth equation above,
dy _du% dx dx'
dc hence, it follows that — = 0. dx
The derivative of a constant is zero.
If the number of functions is infinite, Theorem 4 of Art. 3 may not apply; that is, the limit of the sum may not be equal to the sum of the limits, and hence the derivative of the sum may not be equal to the sum of the derivatives. Thus the derivative of an infinite series cannot always be found by differentiating it term by term.
11. Differentiation of a product.
Let y = uv, wherein u, v are both functions of x.
Then ^=(U + *UW + *V)UV = u^ + v^ + ^ . to,. Ax Ax Ax Ax Ax
Now let Aa; approach zero, using Art. 3, Theorems 4, 5, and
noting that if — has a finite limit, then the limit of Avf—)
Ax \AxJ
is zero. '
The result may be written in the form
d(uv) = udv + vdu (3)
doc dx doc
The derivative of the product of two functions is equal to the sum of the products of the first factor by the derivative of the sec ond, and the second factor by the derivative of the first.
This rule for differentiating a product of two functions may be stated thus : Differentiate the product, treating the first factor as constant, then treating the second factor as constant, and add the two results.
28 DIFFERENTIAL CALCULUS
Cor. To find the derivative of the product of three functions uvw.
Let y = uvw.
By (3), *y = w±(uv)+uv^
dx dx dx
= w(u
dv du\ dw
dx dx ) dx
The result may be written in the form
d(uvw)=uvdw + vwdu + wudv (4
doc dx dm dx
By induction the following rule is at once derived :
The derivative of the product of any finite number of factors is equal to the sum of the products obtained by midtiplying the de rivative of each factor by cdl the other factors.
12. Differentiation of a quotient.
Let y =  , u, v both being functions of x.
Then
u f Au u Au Av
! V U
A?/ v \ Av v _ Ax Ax Ax ~~ Ax v(y + Av)
Passing to the limit, we obtain the result
vduudv
d (u\ dx dx (5)
dx\v J v1
TJie derivative of a fraction, the quotient of two functions, is equal to the denominator multiplied by the derivative of the nu merator minus the numerator multiplied by the derivative of the denominator, divided by the square of the denominator.
DIFFERENTIATION OF THE ELEMENTARY FORMS 29
13. Differentiation of a commensurable power of a function. Let y = un, in which it is a function of x. Then there are three cases to consider :
1. n a positive integer.
2. n a negative integer.
3. n a commensurable fraction.
1. n a positive integer.
This is a particular case of (4), the factors u, v, w, ••• all
being equal. Thus
dy n_xdu
dx dx
2. n a negative integer.
Let n = — m, in which m is a positive integer.
Then y = un = u~m = — ,
«* dl = ^dl by (5), and Case (1)
hence
dx 
u2m 
dx 
— mu~m~ 
idu, dx' 

dy _ dx 
nl dtt wit" — • dx 
3. 7i a commensurable fraction.
Let n=*, where p, g are both integers, which may be either
• q
positive 
or nega 
tive. p 
Then 
y = un = u9 ; 

hence 
if = fir, 

and 

i.e. 
dec cte 
30 DIFFERENTIAL CALCULUS
Solving for the required derivative, we obtain
dx V dxJ
hence *Un = nun14". (6)
dx, doc
The derivative of any commensurable power of a function is equal to the exponent of the power multiplied by the power