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" The differential of a fraction is equal to the denominator into the differential of the numerator, minus the numerator into the differential of the denominator, divided by the square of the denominator. "
Elements of Analytic Geometry and of the Differential and Integral Calculus - Page 129
by Elias Loomis - 1859 - 277 pages
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Elements of the Differential and Integral Calculus

Charles Davies - Calculus - 1836 - 296 pages
...obtain v j. du udv at = 5vv* or by reducing to a common denominator , _ \>du — udv M= ~r— .' hence, the differential of a fraction is equal to the denominator...denominator, divided by the square of the denominator. mm 31 . If the numerator u is constant in the fraction t = — , v its differential will be 0 (Art....
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Elements of the Differential and Integral Calculus

Charles Davies - Calculus - 1838 - 298 pages
...we obtain v dt = 5, du udv vv or by reducing to a common denominator ,. vdu — udv * = p .' hence, the differential of a fraction is equal to the denominator...denominator, divided by the square of the denominator. 31. If the numerator u is constant in the fraction t = —, * 1J its differential will be 0 (Art. 22),...
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Elements of the Differential and Integral Calculus

Albert Ensign Church - Calculus - 1842 - 330 pages
..., or <is sdv du = 3 ; V )T whence by reducing to a common denominator _j* _v^s — *^u /i\ that is, the differential of a fraction is equal to, the denominator...denominator, divided by the square of the denominator. If the denominator be constant, dv = o, and equation (1) becomes vds ds du — —5- = — . VV If...
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Elements of Analytical Geometry and of the Differential and Integral Calculus

Elias Loomis - Calculus - 1851 - 300 pages
...the differential of x(x'+a) (z+26) ? Ans. Ex. 5. What is the differential of ax\xl+a) (x+3b) 1 Ans. PROPOSITION XI. — THEOREM. (184.) The differential...denominator. Let us designate the fraction by -, and suppose 1=9, (0 u=vy. 1 ' Therefore, by Prop. IX., du=ydv+vdy ; whence vdy=du—ydv. ' (2) Substituting in...
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Elements of Analytical Geometry and of the Differential and Integral Calculus

Elias Loomis - Calculus - 1851 - 296 pages
...z(x'+a)(x+2b) ? Ans. Ex. 5. What is the differential of ax'(x'+a) (x+3b) 1 Ans. PROPOSITION XL — THEOREM. (184.) The differential of a fraction is...denominator, divided by the square of the denominator. u Let us designate the fraction by -, and suppose then u=vy. Therefore, by Prop. IX., du=ydv+vdy ;...
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Elements of the Differential and Integral Calculus

Albert Ensign Church - Mathematics - 1855 - 368 pages
...du = --- r; v tr whence by reducing to a common denominator , , A vds — sdv ,.. ч (1), that is, the differential of a fraction is equal to, the denominator into the differential of thc numerator, minus tJ»e numerator into the differential of the denominator, divided by the square...
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Elements of the Differential and Integral Calculus

Albert Ensign Church - Calculus - 1864 - 410 pages
...dl tdv du = --- - ; v v' whence, by reducing to a common denominator, * vds — sdv v (1); that is, the differential of a fraction is equal to the denominator...differential of the numerator, minus the numerator into tht differential of the denominator, divided by the square of the denominator. If the denominator be...
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Elements of the Differential and Integral Calculus

Albert Ensign Church - Calculus - 1870 - 392 pages
...= j-; VV* whence, by reducing to a common denominator, vds — »dv du - d- = v •0); that is, tho differential of a fraction is equal to the denominator...denominator, divided by the square of the denominator. If the denominator be constant, dv = 0, and equation (1) become« or If the numerator be constant,...
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A General Geometry and Calculus: Including Part I. of the General Geometry ...

Edward Olney - Calculus - 1871 - 394 pages
...constant numerator and a variaUe denominator is the product of the numerator with ifa s:gn changed into the differential of the denominator, divided by the square of the denominator. SCH.— If the numerator is variable and the denominator constant it falls tinder Rule 2. 56. RULE...
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Elements of Analytical Geometry, and of the Differential and Integral Calculus

Gerardus Beekman Docharty - Calculus - 1865 - 328 pages
...equal to the sum of the products of the differential of each by the product of all the others. 46. The differential of a fraction is equal to the denominator...denominator divided by the square of the denominator. f(x) Suppose "=7>)' and let x become x+h. «' „,,, - , and u -u= f'(x\f(x+h)-f(x)f(x+h) = " f'(x)f'(x+h)...
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