| Fig. 2. |
4. In the problem of tangents the new process may be described as follows. Let P, P′ be two points of a curve (see fig. 2). Let x, y be the coordinates of P, and x + Δx, y + Δy those of P′. The symbol Δx means “the difference of two Differentiation. x’s” and there is a like meaning for the symbol Δy. The fraction Δy/Δx is the trigonometrical tangent of the angle which the secant PP′ makes with the axis of x. Now let Δx be continually diminished towards zero, so that P′ continually approaches P. If the curve has a tangent at P the secant PP′ approaches a limiting position (see § 33 below). When this is the case the fraction Δy/Δx tends to a limit, and this limit is the trigonometrical tangent of the angle which the tangent at P to the curve makes with the axis of x. The limit is denoted by
| dy | . |
| dx |
If the equation of the curve is of the form y = ƒ(x) where ƒ is a functional symbol (see [Function]), then
| Δy | = | ƒ(x + Δx) − ƒ(x) | , |
| Δx | Δx |
and
| dy | = lim.Δx = 0 | ƒ(x + Δx) − ƒ(x) | . |
| dx | Δx |
The limit expressed by the right-hand member of this defining equation is often written
ƒ′(x),
and is called the “derived function” of ƒ(x), sometimes the “derivative” or “derivate” of ƒ(x). When the function ƒ(x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ(x) = x2, we have