Page:Calculus Made Easy.pdf/98

the slope of a very small portion of the curve situated just at that point; and we have seen that this is the same as “the slope of the tangent to the curve at that point.”

Observe that $dx$ is a short step to the right, and $dy$ the corresponding short step upwards. These steps must be considered as short as possible–in fact indefinitely short,–though in diagrams we have to represent them by bits that are not infinitesimally small, otherwise they could not be seen.

We shall hereafter make considerable use of this circumstance that ${\tfrac {dy}{dx}}$ represents the slope of the curve at any point.

Fig. 8.

If a curve is sloping up at $45^{\circ }$ at a particular point, as in Fig. 8, $dy$ and $dx$ will be equal, and the value of ${\frac {dy}{dx}}=1$.