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Of the differentiation of algebraic quantities.
1. We say that one variable is a function of another when the first is equal to a certain compound analytical expression of the second; for example, y is a function of x in the following equations: y = (a2 - x2)½, y = x3- 3bx2, y = x4/a , y = l + cx2 , y = (a + bx + cx2 + dx3)m/n.
2. Let us consider a function in its state of augmentation, in consequence of the increase of the variable which it contains. As every function of a variable x can be represented by the ordinate of a curve BMM', let AP = x and PM = y be the ordinates of a point M of this curve, and let us suppose that the abscissa AP receives an increment PP' = h; the ordinate PM will become P'M' = y'. Fig. 1. To obtain the value of this new ordinate, we see that it is necessary to change x into x + h, in the equation of the curve, and the value which this equation will then determine for y will that of y'.
For example, if we had the equation y = mx y' = m(x + h)2, we should obtain y' by changing x into x + h, and y into y', and we would have2 = mx2 + 2mxh + mh2
3. Let us now take the equation y = x y' = (x + h) y [y'/h - y/h] = 3x3 . . . . . (1),
and let us suppose that y becomes y' when x becomes x + h; we shall have
if from this equation we subtract equation (1), there will remain3 = x3 + 3x2h + 3xh2 + h3:1 - y = 3x2h + 3xh2 + h3; and dividing by h2 + 3xh + h2 . . . . . (2).
Let us see what this result teaches us: y' - y represents the increment of the function y in consequence of the increment h given to x, since this difference y' - y is that of the new state of magnitude of y, as respects its primitive state.
On the other hand the increment of x being h, it follows that [y'/h - y/h] is the ratio of the increment [...]