...Uh?, Mi — U A+ Uh; h but M, — и is the increment of u, and h is the increment of ¿v ; therefore the ratio of the increment of the function, to the increment of x, = A + Uh ; and as h decreases, this ratio tends to A as its limit, and when h vanishes actually...

...I—a 2x Answers. (°~1^, , - ^ - r, 2(l+x)h+h*. x(x+h)' (a+x+h) (a+x)' v T ' 6. Required the limiting ratio of the increment of the function to the increment of the variable, in the following functions, 2a2, x3-ax, 5x*+a, 3x*+c, x3+x*-ax+c, x+a 3 ~x~> X~x Answers. 4x, 3x*—a,...

...If у be a fonction of x, the ßrtt differential co-efficient off may bt cotuidertd ai ihr limit of the ratio of the increment of the function to the increment of Л* triable. In ill extensive treatise* upon the differential calculai, the manner in which '.'if differential...

...u^—u-\-Ah-\- U№, .. я but M, — M is the increment of щ and A is the increment of x ; therefore the ratio of the increment of the function, to the increment of x, —A+Uh; and as A decreases, this ratio tends to A as its limit, and when я vanishes actually =...

...MI = м + A h + UJf, hut M! — « is the increment of «, and h is the increment of x ; therefore the ratio of the increment of the function to the increment of x = A -f- U h ; and as A decreases this ratio tends to A as its limit and becomes — A, whon A vanishes....

...given, and if the corresponding increment of a function of the variable be determined, then the limit of the ratio of the increment of the function to the increment of the variable, when the increment of the variable is taken indefinitely small, is the derivative of the function as...

...x = x1 ; then у — у' + Ay, when x = x' + A x. 237. The Derivative of a function is the limit of the ratio of the increment of the function to the increment of the variable as the increment of the variable approaches zero as its limit. If у is a function of x, the derivative...

...ф(x), Дa; /t -^ = limit of yi^iJ— — 2ií¿, as Л approaches zero. dx h as the limiting value of the ratio of the increment of the function to the increment of the variable, as these increments approach zero. That is, the differential coefficient of the function ф(x) with...

...zero. dx h The differential coefficient of a function may then be defined as the limiting value of the ratio of the increment of the function to the increment of t)ie variable, as these increments approach zero. That is, the differential coefficient of the function...

............ ••••More concisely, the derived function (when it exists) is the "limiting value of the ratio of the increment of the function to the increment of the independent variable, when both increments are indefinitely diminished. It is to be carefully noticed...