An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms |
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Page 111
... x2 + & c . = 0 ; whence , by the preceding theorem , that is , A - A ' 0 , B - B ' = 0 , C - C = 0 , & c .; A A ' , BB ' , C = C " , & c . A Function ; its Variable , and Rate of Change CH . IV . I. ] 111 NUMERICAL EQUATIONS .
... x2 + & c . = 0 ; whence , by the preceding theorem , that is , A - A ' 0 , B - B ' = 0 , C - C = 0 , & c .; A A ' , BB ' , C = C " , & c . A Function ; its Variable , and Rate of Change CH . IV . I. ] 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 ' —v , and therefore the derivative of the sum is u ' -U v + 2 which is obviously the sum of their derivatives . 169. Corollary . By ...
... variable , be u ' and v ' ; the increase of their sum will be or ( u ' + v ' ) — ( u + v ) u ' — u + v ' —v , and therefore the derivative of the sum is u ' -U v + 2 which is obviously the sum of their derivatives . 169. Corollary . By ...
Page 114
... variable . To find the derivative of any 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 b - an • - a Now when bis equal to a , the value of this quotient ...
... variable . To find the derivative of any 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 b - an • - a Now when bis equal to a , the value of this quotient ...
Page 115
... variable . 1. x2 . 2. x3 . Ans . 2 x . Ans . 3 x2 . 3. xn + a xm + b x2 + & c . Ans . n xn − 1 + m a x TM -1 + p b ... variable , it is evident The Derivative of a Product . that when the variable CH . IV . 115 NUMERICAL EQUATIONS .
... variable . 1. x2 . 2. x3 . Ans . 2 x . Ans . 3 x2 . 3. xn + a xm + b x2 + & c . Ans . n xn − 1 + m a x TM -1 + p b ... variable , it is evident The Derivative of a Product . that when the variable CH . IV . 115 NUMERICAL EQUATIONS .
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126 become zero 3d root arithmetical progression coefficient commensurable roots common difference contained continued fraction continued product Corollary deficient terms denote derivative Divide dividend division equal roots equal to zero 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