A Treatise on the Differential Calculus |
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Page 6
... dy dx or In this expression dx and dy are any quantities whatever , either finite or infinitesimal , which are in the ratio of the dy ultimate values of 8x and dy . The fraction is called the dx differential coefficient of y with regard ...
... dy dx or In this expression dx and dy are any quantities whatever , either finite or infinitesimal , which are in the ratio of the dy ultimate values of 8x and dy . The fraction is called the dx differential coefficient of y with regard ...
Page 7
... dy dx ' and the right - hand member assumes its limiting value 3x2 : thus or dy dx = 3x2 , dy = 3x2 dx , that is , the differential coefficient of x with respect to x is 3x2 , and its differential is 3x2dx . Differentiation of a ...
... dy dx ' and the right - hand member assumes its limiting value 3x2 : thus or dy dx = 3x2 , dy = 3x2 dx , that is , the differential coefficient of x with respect to x is 3x2 , and its differential is 3x2dx . Differentiation of a ...
Page 9
... dy dy dys == + + dx dx dx dx dy . dx ... In fact , taking any simultaneous values x ' , u ' , y ' ,, y ' ,, Y's , ••• Y ' „ › of x , u , y1 y2 y3 , whence u ' U x = or x ' - ... u = y , we have n Y1 + y 2 + Y 3 u ' = y'1 + Y'2 + Y's + + ...
... dy dy dys == + + dx dx dx dx dy . dx ... In fact , taking any simultaneous values x ' , u ' , y ' ,, y ' ,, Y's , ••• Y ' „ › of x , u , y1 y2 y3 , whence u ' U x = or x ' - ... u = y , we have n Y1 + y 2 + Y 3 u ' = y'1 + Y'2 + Y's + + ...
Page 10
... dy , and dy , each equal to zero , we have or du dx du 1 = = dy , dy 2 + y 2 dx Y1 dx y1 dy2 + y2 dy1 . 9 Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied ...
... dy , and dy , each equal to zero , we have or du dx du 1 = = dy , dy 2 + y 2 dx Y1 dx y1 dy2 + y2 dy1 . 9 Hence the differential coefficient of the product of two functions is equal to the sum of the products of each function multiplied ...
Page 11
... dy , dx Y1 dy Y1 + - Y2 dx + 1 dy2 Y3 dx 1 dy Y " 1 dyn dx dy3 dy , + dy Y2 Y3 · + Yn Relation between Inverse Differential Coefficients . 18. If y , be a function of x , in which case x will also be then a function of y , dy dx • dx dy ...
... dy , dx Y1 dy Y1 + - Y2 dx + 1 dy2 Y3 dx 1 dy Y " 1 dyn dx dy3 dy , + dy Y2 Y3 · + Yn Relation between Inverse Differential Coefficients . 18. If y , be a function of x , in which case x will also be then a function of y , dy dx • dx dy ...
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Common terms and phrases
algebraical arbitrary functions asymptote axis Cambridge change sign constant cosec curve d'u dy d²r d²u d²x d²z d2z d2z d³u d³x d³y d³z denote df df df dx DIFFERENTIAL CALCULUS differential equations du du dy du² dv₁ dx dx dx dy dx dx dz dx² dx³ dxdy dy dF dy dx dy dy dy dy dz dy₁ dy₂ dy³ dz dx dz dy dz dz eliminate expression f(y₁ find the Differential formula ƒ Y₁ Hence implicit function increment independent variables indeterminate limit maxima and minima maximum or minimum minimum value multiplying negative partial differential coefficients points of inflection positive quantity putting regard shews Suppose tangent Taylor's Theorem total differential Trinity College University of Cambridge whence y+dy Y₁ Y₂ zero бу бх
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