A Treatise on the Differential Calculus |
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Page 38
... proposed to find given that dx in terms of x , having dy y = sin x . By Art . ( 33 ) , we have dy = COS X , dx and , by Art . ( 18 ) , dy dx 1 : dx dy dx hence COS X • 1 , = sec x . dy dx dy = Ex . 9. If u = y3 , and y = cot x ; let it ...
... proposed to find given that dx in terms of x , having dy y = sin x . By Art . ( 33 ) , we have dy = COS X , dx and , by Art . ( 18 ) , dy dx 1 : dx dy dx hence COS X • 1 , = sec x . dy dx dy = Ex . 9. If u = y3 , and y = cot x ; let it ...
Page 41
... proposed to find z = sin ( xy ) , du du du Du Du Du . dx ' dy dy ' dz dx ' dy du Xyz By Art . ( 30 ) , = yz . dx du by Art . ( 45 ) , - dy du and = log , x dz X log , x.x12 . • d ( yz ) xyz . dz = y log , x d ( yz ) = z . log , x . x ...
... proposed to find z = sin ( xy ) , du du du Du Du Du . dx ' dy dy ' dz dx ' dy du Xyz By Art . ( 30 ) , = yz . dx du by Art . ( 45 ) , - dy du and = log , x dz X log , x.x12 . • d ( yz ) xyz . dz = y log , x d ( yz ) = z . log , x . x ...
Page 50
... proposed equation to the form , Order of Partial Differentiations indifferent . 48. The following is a theorem of great importance in suc- cessive differentiation if u = f ( y1 , Y2 ) , then ddu = dy dv2u . For and therefore - §11u = ƒ ...
... proposed equation to the form , Order of Partial Differentiations indifferent . 48. The following is a theorem of great importance in suc- cessive differentiation if u = f ( y1 , Y2 ) , then ddu = dy dv2u . For and therefore - §11u = ƒ ...
Page 60
... proposed to change the variables of an equation dy d'y d3y ƒ x , y , dx ' dx2 ' dx3 ) = = 0 . . ( 1 ) from x and y into two variables s and t , t being , in the trans- formed equation , and x in the proposed equation , the inde- pendent ...
... proposed to change the variables of an equation dy d'y d3y ƒ x , y , dx ' dx2 ' dx3 ) = = 0 . . ( 1 ) from x and y into two variables s and t , t being , in the trans- formed equation , and x in the proposed equation , the inde- pendent ...
Page 61
... of independent Variables into another . 54. Let z be a function of two independent variables x and y . We propose to express the partial differential coef- ficients of z , taken with regard to x and SUCCESSIVE DIFFERENTIATION . 61.
... of independent Variables into another . 54. Let z be a function of two independent variables x and y . We propose to express the partial differential coef- ficients of z , taken with regard to x and SUCCESSIVE DIFFERENTIATION . 61.
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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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