An Elementary Treatise on Algebra |
From inside the book
Results 1-5 of 11
Page 111
... 0 ; whence , by the preceding theorem , that is , A — A ′ = 0 , В — В ′ = 0 , C - C ' = 0 , & c . ; A = A ' , B = B ' , C = C ' , & c . A Function ; its Variable , and Rate of Change CH . IV . § 1. ] 111 NUMERICAL EQUATIONS .
... 0 ; whence , by the preceding theorem , that is , A — A ′ = 0 , В — В ′ = 0 , C - C ' = 0 , & c . ; A = A ' , B = B ' , C = C ' , & c . A Function ; its Variable , and Rate of Change CH . IV . § 1. ] 111 NUMERICAL EQUATIONS .
Page 112
... variables , the rela- tive rate of change of the function and the variable , that is , the ratio of the change in the value of the function to that in the value of the variable , is called the derivative of the function . The derivative ...
... variables , the rela- tive rate of change of the function and the variable , that is , the ratio of the change in the value of the function to that in the value of the variable , is called the derivative of the function . The derivative ...
Page 113
... variable , be u ' and v ' ; the increase of their sum will be or ( u ' + v ' ) - ( u + v ) u ' - u + v ' υ , and therefore the derivative of the sum is u ' i 2 - + - 0 , which is obviously the sum of their derivatives . 169. Corollary ...
... variable , be u ' and v ' ; the increase of their sum will be or ( u ' + v ' ) - ( u + v ) u ' - u + v ' υ , and therefore the derivative of the sum is u ' i 2 - + - 0 , which is obviously the sum of their derivatives . 169. Corollary ...
Page 114
... variable . Solution . Let the variable be a and the power a " , and let 6 differ infinitely little from a ; the derivative of an is then bn an b a Now when b is equal to a , the value of this quotient is , nan - 1 ; by art . 51 , and ...
... variable . Solution . Let the variable be a and the power a " , and let 6 differ infinitely little from a ; the derivative of an is then bn an b a Now when b is equal to a , the value of this quotient is , nan - 1 ; by art . 51 , and ...
Page 115
... variable . 1. x2 . 2. x3 . 3. xn + a xm + b xp + & c . Ans . 2 x . Ans . 3 x2 . Ans . nxn - 1 + maxm - 1 + pbxp - 1 ... variable , it is evident The Derivative of a Product . that when the variable CH . IV . § II . ] 115 NUMERICAL ...
... variable . 1. x2 . 2. x3 . 3. xn + a xm + b xp + & c . Ans . 2 x . Ans . 3 x2 . Ans . nxn - 1 + maxm - 1 + pbxp - 1 ... variable , it is evident The Derivative of a Product . that when the variable CH . IV . § II . ] 115 NUMERICAL ...
Other editions - View all
Common terms and phrases
126 become zero 3d root arithmetical mean arithmetical progression Binomial Theorem coefficient commensurable roots common difference contained continued fraction continued product Corollary deficient terms denote derivative Divide dividend division equal roots equation x² factor Find the 3d Find the 4th Find the continued Find the greatest Find the number Find the square Find the sum Free the equation Geometrical Progression given equation given number gives greatest common divisor Hence imaginary roots last term least common multiple letter logarithm monomials multiplied number of real number of terms polynomial positive roots preceding article Problem quantities in example quotient radical quantities ratio real roots reduced remainder required equation required number row of signs Scholium Second Degree Solution Solve the equation square root Sturm's Theorem subtracted Theorem unity unknown quan unknown quantity variable whence у³
Popular passages
Page 48 - In any proportion the terms are in proportion by Composition and Division ; that is, the sum of the first two terms is to their difference, as the sum of the last two terms is to their difference.
Page 127 - Multiply the divisor, thus increased, by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend.
Page 55 - There is a number consisting of two digits, the second of which is greater than the first, and if the number be divided by the sum of its digits, the quotient is 4...
Page 192 - One hundred stones being placed on the ground in a straight line, at the distance of 2 yards from each other, how far will a person travel who shall bring them one by one to a basket, placed at 2 yards from the first stone ? Ans.
Page 63 - A term may be transposed from one member of an equation to the other by changing its sign.
Page 1 - ALGEBRA. CHAPTER I. FUNDAMENTAL PROCESSES OF ALGEBRA. SECTION I. Definitions and Notation. 1. Algebra, according to the usual definition, is that branch of mathematics in which the quantities considered are represented by the letters of the alphabet, and the operations to be performed upon them are indicated by signs. In this sense it would embrace almost the whole science of mathematics, elementary geometry alone being excepted. It is, consequently, subject in common use to some limitations, which...
Page 113 - The derivative of the sum of two functions is the sum of their derivatives.
Page 127 - Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend. III. Double the root already found and place it on the left for a divisor.
Page 271 - The characteristic of the logarithm of a decimal fraction is a negative number* and is equal to the number of places by which its first significant figure is removed from the place of units. Thus the logarithm of...
Page 196 - Hence, to find the sum, multiply the first term by the difference between unity and that power of the ratio whose exponent is equal to the number of terms, and divide the product by the difference between unity and the ratio. Examples in Geometrical Progression. 260. Corollary. The two equations...