SECTION 3.-Examples of the infinitesimal method.

24. Illustrative examples on differentiation

CHAPTER II.

CONSTRUCTION OF RULES FOR DIFFERENTIATION OF FUNCTIONS.

SECTION 1.-The differentiation of explicit functions of one variable.
25-29. Rules for differentiating

40

f(x)

f(x)±c, cƒ (x), ƒ (x)±$(x) ± . ., ƒ (x) × $ (x),

(x)

30. Differentiation of a"....

31. Differentiation of compound functions

32-36. Differentiation of a*, ex, logax, logex

37. Differentiation of a product of many functions

38-45. Differentiation of circular functions....

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

46. Functions of many variables ..

47. Differentiation of a function of two independent variables

48. Differentiation of an implicit function of two variables. . . . . .

49, 50. Differentiation of a function of many independent variables
51. Differentiation of functions of many variables, all of which are
not independent

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

variables enter in particular combinations

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..

55. Leibnitz's Theorem for successive differentiation

56. The nth differential of f (x), in terms of successive values

of f (x)

SECTION 2.-Expansion of an explicit function of one variable.

57. Maclaurin's Theorem...

58. Limits of Maclaurin's Theorem.

59. Examples of Maclaurin's Theorem

60-63. Certain trigonometrical expressions and theorems

64-66. The roots of +1 and of — 1 ..

---

67. Imaginary logarithms

103

108

113

SECTION 3.-Theory of the equicrescent variable, and Taylor's Theorem.

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

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

times

69, 70. Modification of the preceding, when r is equicrescent

71. Taylor's Series....

72. Limits of Taylor's Series

73. Examples of Taylor's Series

74. Expansion of f(x) in ascending powers of x-a

115

116

119

119

121

123

SECTION 4.-Change of equicrescent variable, and transformation of

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

83. Transformation of partial derived functions..

137

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 formulæ

142

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

terms of the other by means of Maclaurin's Theorem

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

92. Expansion of f(x) in ascending powers of (x)

93. Laplace's Theorem

94. Extension of Maclaurin's Theorem

95. Arbogast's process of derivation.

96. Extension to implicit functions

97. Extension to functions of many series..

155

156

157

158

162

163

SECTION 7.-The formation of differential equations by means of

elimination.

98. Elimination of constants from an explicit function

99. Elimination of constants from an implicit function

100. Elimination of given functions

171

172

174

101. Trigonometrical relations expressed by differential equations

102. Formation of differential equations in terms of three and

more variables

103-105. Formation of differential equations by the elimination

of arbitrary functions..

SECTION 8.-Transformation of partial differential expressions.

106. Transformation of expressions involving partial derived-

functions into their equivalents in terms of other variables

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

CERTAIN RELATIONS BETWEEN FUNCTIONS AND DERIVED-FUNCTIONS.

109. Extension of preceding principles to the consideration of

f(x), when x receives a finite increment

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

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

creases, and vice versá

187

188

111. The proof that

F(x) — F(x) = (xn−xo) F' {xo + 0 (x − xo) },

-

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

tween x and xo

112. The proof that under stated conditions

=

F(xo+h) F(xo) F'(xo + Oh)
f(xo+h) f(xo) f'(xo+Oh)

113. Hence also under certain conditions

F(xo+h)-F (xo) F" (x+0h)

190

191

193

114. If in the theorem of the last Article ƒ(x) =

=

(x-x)", then

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

116. Particular cases of the theorem of Art. 114

CHAPTER V.

THE ORDERS OF INFINITESIMALS, AND EVALUATION OF

INDETERMINATE QUANTITIES.

117. Unity accurately defined

195

195

196

198

SECTION 1.-Orders of infinitesimals.

118. Definition of order of infinitesimals

119. Mode of determining the order of an infinitesimal

120. Examples wherein orders of infinitesimals are determined

SECTION 2.-Evaluation of quantities of the forms

0 ∞ 0 ×∞, ∞ 1 ∞

121. The cause why quantities assume the forms and

122-124. Modes of evaluating quantities of the form

125. Evaluation of quantities of the form

126. Evaluation of quantities of the form 0 × ∞

127. Evaluation of quantities of the form ∞

SECTION 3.-Evaluation of quantities of the forms 0o, ∞o, 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

133. Another form of Taylor's Theorem

215

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.

с

SECTION 2.-Functions of many variables.

137. The theorem for functions of two variables analogous to

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 ƒ (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

independent variables.

154. Method of determining such values, and examples

250

......

SECTION 3.-Maxima and minima of an explicit function of two

independent variables.

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

ables

252

156, 157. Method of determining criteria of such singular values

158. Another method of determining such singular values

253

255