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Section 3.—Examples of the infinitesimal method.

24. Illustrative examples on differentiation 40

CHAPTER II.

CONSTRUCTION OF RULES FOR DIFFERENTIATION OF FUNCTIONS.

Sbction 1.—The differentiation of explicit functions of one variable.

25-29. Rules for differentiating

/(*) ± c, c/(x). f(x) + <b(x) ± • •. /(*) x <p (x), ^ . 51

30. Differentiation of xn 54

31. Differentiation of compound functions 56

32-3(5. Differentiation of a*, e*, logo*, log, J 57

3/. Differentiation of a product of many functions 60

38-45. Differentiation of circular functions 61

Section 2.—The differentiation of functions of many variables.

46. Functions of many variables 70

47. Differentiation of a function of two independent variables ... 73

48. Differentiation of an implicit function of two variables 75

49. 50. Differentiation of a function of many independent variables 77

51. Differentiation of functions of many variables, all of which are

not independent 79

52, 53. Differentiation of functions of many variables, when the

variables enter in particular combinations 81

CHAPTER III.

SUCCESSIVE DIFFERENTIATION.

Section 1.—Successive differentiation of an explicit function of one

, variable.

54. Explanation of the process, and corresponding nomenclature

and symbols. Examples 89

55. Leibnitz's Theorem for successive differentiation 94

56. The nth differential of / (j), in terms of successive values

of f(x) 97

Section 2.—Expansion of an explicit function of one variable.

57. Maclaurin's Theorem 97

58. Limits of Maclaurin's Theorem 99

59. Examples of Maclaurin's Theorem KM) CO—63. Certain trigonometrical expressions and theorems 103

54-66. The roots of + 1 and of — 1 108

67. Imaginary logarithms 113

Section 3.—Theory of the equicrescent variable, and Taylor's Theorem.

68. The relation between y and its equivalent/(x), of the equation

y = f (x). when x has been successively increased n

times 115

69. 70. Modification of the preceding, when * is equicrescent .. 116

71. Taylor's Series 119

72. Limits of Taylor's Series 119

73. Examples of Taylor's Series 121

74. Expansion of /(*) in ascending powers of x—a 123

Section 4.—Change of equicrescent variable, and transformation of

differential expressions.

75. Different forms of the problem 123

76. Requisite formulae for a function of two variables 124

77- Transformations in terms of a new variable 125

Section 5.—Successive differentiation of functions of many

independent variables.

78. Explanation of the symbols 128

79. The order of successive differentiations with respect to many

variables is indifferent 131

80. 81. Application of the principles of the preceding Articles to

functions of two and more variables 133

82. Euler's Theorems of homogeneous functions 137

83. Transformation of partial derived functions 139

Section 6.—Successive differentiation of implicit functions.

84. Calculation of derived-functions when the relation between

the variables is given in an implicit form .. 141

85. Examples of preceding formula? 142

86. Expansion of one of the variables of an implicit function in

terms of the other by means of Maclaurin's Theorem . . 143

87- Calculation and properties of Bernoulli's numbers 145

88. Examples of Bernoulli's numbers 146

89. The sums of the powers of the reciprocals of the natural

numbers 148

90. Lagrange's Theorem 149

91. Examples of Lagrange's Theorem 153 92. Expansion of f(x) in ascending powers of <f>(x) 135 114. If in the theorem of the last Article J\x) = (x— x,i)", then

93. Laplace's Theorem ,. 150

94. Extension of Maclaurin's Theorem 157

95. Arbogast's process of derivation 158

96. Extension to implicit functions 162

97- Extension to functions of many series 163

Sbction 7-—The formation of differential equations by means of

elimination.

98. Elimination of constants from an explicit function 165

99. Elimination of constants from an implicit function 167

100. Elimination of given functions 169

101. Trigonometrical relations expressed by differential equations 171

102. Formation of differential equations in terms of three and

more variables 172

103-105. Formation of differential equations by the elimination

of arbitrary functions 174

Section 8.—Transformation of partial differential expressions.

106. Transformation of expressions involving partial derived-

functions into their equivalents in terms of other variables 180

107, 108. Examples illustrative of the preceding principles .... 182

CHAPTER IV.

CERTAIN RELATIONS BETWEEN FUNCTIONS AND DERIVED-FUNCTIONS.

109. Extension of preceding principles to the consideration of

f(x), when x receives a finite increment 187

110. According as J'{x) is positive or negative, so does f(x) in-

crease and diminish with x, or as x increases, f(x) de-

creases, and vice versd 188

111. The proof that

* (*n) - » fco) = (x„ — v {x0 + 6 (xn — x0)},

F (x) being finite and continuous for all values of x be-

tween x„ and xo 190

w(xo + h)-r{x») = —- »"(*&+**) ... 195

115. Certain corollaries of the theorem of Art. 114 195

1 16. Particular cases of the theorem of Art. 114 196

CHAPTER V.

THE ORDERS OF INFINITESIMALS, AND EVALUATION OF

INDETERMINATE QUANTITIES.

II/. Unity accurately defined 198

Section 1.—Orders of infinitesimals.

118. Definition of order of infinitesimals 199

119. Mode of determining the order of an infinitesimal 200

120. Examples wherein orders of infinitesimals are determined 201

Section 2.—Evaluation of quantities of the forms

JJ. ~. Oxoc, oo-oo.

121. The cause why quantities assume the forms ^ and .. 202

122-124. Modes of evaluating quantities of the form ..... 202

125. Evaluation of quantities of the form ||- 206

126. Evaluation of quantities of the form Oxoo 209

127- Evaluation of quantities of the form 00 — oo 209

Section 3.—Evaluation of quantities of the forms 0°, oo°, 1 " , 0*.

128. Mode of evaluating, and examples of, such quantities .... 210

CHAPTER VI.

EXPANSION OF FUNCTIONS.

Section 1.—Functions of one variable.

129. An accurate proof of Taylor's Series 212

130. The imperfect form of it given in Art. 71 214

131. Deduction of Maclaurin's Theorem from Taylor's 214

132. The imperfect form of Maclaurin's Theorem given in Art. 57 215

133. Another form of Taylor's Theorem 215

134. 135. The limits of Taylor's and Maclaurin's Series 215

136. The failure of Taylor's and Maclaurin's Series 217

PRICE, VOL. I. c

Section 2.—Functions of many variables.

137- The theorem for functions of two variables analogous to

that of Art. 111 222

138. A particular form of the preceding 224

139. Evaluation of indeterminate quantities which are functions

of two independent variables 225

140. Expansion of F(x + h, y + k) 228

141. Examples of the preceding 229

142. Expansion of F (x+h, y + A, z +I ) 230

143. Extension of the process of derivation to functions of two

and more variables 233

CHAPTER VII.

MAXIMA AND MINIMA.

144. Definition of maximum and minimum 235

Section 1.—Maxima and minima of explicit functions of one variable.

145. Method of determining such singular values of f(x) 236

146. Easy detection of the singular values when f(x) is alge-

braical 237

147- Geometrical representation of the criteria 238

148. Examples illustrative of the preceding methods 238

149, 150. The forms of the derived-functions of f(x), when it

has such singular values; and a method of thereby deter-

mining maxima and minima of f(x) 240

151. Examples of maxima and minima 243

152. The determination of the number of maxima and minima of

a given function 247

153. The absolute maximum and minimum 249

Section 2.—Maxima and minima of implicit functions of two

154. Method of determining such values, and examples 250

Section 3.—Maxima and minima of an explicit function of two

indejtendent variables.

155. Definition of maxima and minima of functions of two vari-

ables 252

156. 157- Method of determining criteria of such singular values 253

158. Another method of determining such singular values .... 255