A course of pure mathematics, by G.H. Hardy.

CHAPTER VI.

DERIVATIVES AND INTEGRALS
(A FIRST CHAPTER IN THE DIFFERENTIAL AND INTEGRAL CALCULUS).
91. Derivatives or Differential Coefficients. Let us return
to the consideration of the properties which we naturally associate
with the notion of a curve. The first and most obvious property
is, as we saw in the last chapter, that which gives a curve its
appearance of connectedness, and which we embodied in our definition of a continuous function, i.e. a function whose graph is a
continuous curve, such as the curve C of Fig. 38.
The ordinary curves which occur in elementary geometry, such
as straight lines (which we have agreed, for convenience, to class
as curves), circles and conic sections, have of course many other
properties of a general character. The simplest and most noteworthy of these is perhaps that they have a definite direction at
every point, or what is the same thing, that at every point of the
curve we can draw a tangent to it. The reader will probably
remember that in elementary geometry the tangent to a curve at
P is defined to be 'the limiting position of the chord PQ, when Q
moves up towards coincidence with P.' Let us consider what is
implied in the assumption of the existence of such a limiting
position.
In the figure (Fig. 42) P is a fixed point on the curve, and Q
a variable point; PM, QNr are parallel to O Y and PR to OX.
We can denote the coordinates of P by x, y and those of Q by
x + h, y + k: Ah will be positive or negative according as N lies to
the right or left of M.
We have assumed that there is a tangent to the curve at P,
or that there is a definite 'limiting position' of the chord PQ.
Suppose that the tangent at P, PT, makes an angle r with OX.