An Elementary Treatise on Algebra: To which are Added Exponential Equations and Logarithms |
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Page 111
... C ' ) 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 .
... C ' ) 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
... function of the a , b , and x ; but if x is variable while a and b are constant , it is more usual to regard y as simply a function of x . 165. Definition . In the case of a change in the value of a function , arising from an infinitely ...
... function of the a , b , and x ; but if x is variable while a and b are constant , it is more usual to regard y as simply a function of x . 165. Definition . In the case of a change in the value of a function , arising from an infinitely ...
Page 113
... Functions . 168. Theorem . The derivative of the sum of two functions is the sum of their derivatives . Proof . Let the two ... function is multiplied by a constant its deriva- tive must be multiplied by the same constant . Thus , if the ...
... Functions . 168. Theorem . The derivative of the sum of two functions is the sum of their derivatives . Proof . Let the two ... function is multiplied by a constant its deriva- tive must be multiplied by the same constant . Thus , if the ...
Page 114
... function . Solution . Let the variable be a , the function u , and the power u " ; let b differ infinitely little from a , and let v be the corresponding value of u ; if U is the derivative of u and U that of u ” , we have vn un v น σ ...
... function . Solution . Let the variable be a , the function u , and the power u " ; let b differ infinitely little from a , and let v be the corresponding value of u ; if U is the derivative of u and U that of u ” , we have vn un v น σ ...
Page 115
... function , and diminishing the exponent by unity . 175. EXAMPLES . Find the derivatives of the following functions ... function to that of the variable , it is evident The Derivative of a Product . that when the variable CH . IV . 115 ...
... function , and diminishing the exponent by unity . 175. EXAMPLES . Find the derivatives of the following functions ... function to that of the variable , it is evident The Derivative of a Product . that when the variable CH . IV . 115 ...
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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