A Treatise On The Differential And Integral Calculus, And The Calculus Of Variations - Tapa dura

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By a method, similar to that of M. Poisson, it is shewn that Ui =f{x +h) can always be expanded in a series of ascending integral powers of h, which may be written under the convenient form of the equation M, =u +A h+ Ulr. The term A h, the first term of the difference between u-u, is defined to be the differential of u: and A the coefficient of his called the differential coefficient: and from these definitions, the rules forD ifferentiation are in general derived. But since when the general form off{x +h) has been demonstrated, we see that -is equal to A, when h= 0: we may therefore find the differential coefficient, by dividing both sides of the equation u-u (p (x, h) (not expanded), by h, and then make h= 0. This method which sometimes diminishes the algebraical labour of finding A, is in few instances made use of. It is in fact the method of Limits, often useful in the application of the Calculus, and to which the idea of series is so necessary an auxiliary. In truth the notion of a series seems inseparably connected with the method of Limits to which in theD iflferential Calculus it gives a clearness and precision, of which it stands much in need. For to say that when n=f{x); and nis its value when xbecomes xthe differential Coefficient is the value to which ~tends x- Xwhile xcontinually approaches x, without first exhibiting the relation which exists between ?, and ,is to use a mysterious obscurity which must heavily tax the faith or the credulousness of the reader. But put for 7i its expansion and the difficulty vanishes. It has been said that(.
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