THE BEGINNINGS OF DIFFERENTIATION


This was motivated in the early 17th century by four types of problem :

  1. Given a formula for distance as a function of time, d = f(t), find the velocity and acceleration at any given instant.

  2. Given a curve, find the tangent (or normal) to the curve at a given point.

  3. Find the maximum or minimum value of a function.

  4. Find the length of a given curve.

Early progress on these problems (the first steps in using differentiation) were by Fermat (1601-1665) and Barrow (1630-1677).

Example of Barrow's Work

Given a curve and a point P on it, find N on the x-axis so that PN is a tangent to the curve.
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Barrow takes another point P' on the curve, and forms PRQ and PMN , which are similar, hence the slope = .

Then when the arc PP' is sufficiently small he identifies it with the line PQ (lack of rigour again !)
In one of his calculations he deals with the curve y = kx (k constant), replaces x by x+e and y by y+a to get

y + 2ay + a = kx + ke

He subtracts y = kx to get

2ay + a = ke

The identification of PP/ with PQ is equivalent to discarding nonlinear terms in a and e

(a here), so he concludes 2ay = ke, i.e. = .

Then, since = = he has = .

Since PM is y, he has calculated MN, and hence located the position of N.

With the state of the calculus reaching this point, the stage was set for Newton and Leibniz to develop it into a form recognisable by students today.


Assessment

The Development of Calculus

History of Mathematics Module

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