## Abstract

Returning 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 \(\frac{{{d^2}y}}{{d{x^2}}}\) mean in this case? Clearly it means the rate (per unit of length *x*) at which the slope is changing—in brief, it is *an indication of the manner in which the slope of the portion of curve considered varies*, that is, whether the slope of the curve increases or decreases when *x* increases, or, in other words, whether the curve curves up or down towards the right.

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