Page:Calculus Made Easy.pdf/132



to the process of successive differentiation, it may be asked: Why does anybody want to differentiate twice over? We know that when the variable quantities are space and time, by differentiating twice over we get the acceleration of a moving body, and that in the geometrical interpretation,

as applied to curves, $$\frac {dy}{dx}$$ means the slope of the curve. But what can $$\frac {d^2y}{dx^2}$$ mean in this case? Clearly it means the rate (per unit of length $$x$$) at which the slope is changing—in brief, it is a measure of the curvature of the slope.

Suppose a slope constant, as in Fig. 31.

Here, $$\frac {dy}{dx}$$ is of constant value.