What is a derivative?
The derivative of a function f(x) is a measure of how the function f changes as its variable x changes. You have already met an example of derivatives: the gradient, m, measuring the rate of change of f(x) and x.
Indeed, to find m, you start by considering two points x1 and x2. You find their difference, Delta x:
Delta x = x2 – x1.
Then, you find the value of the function f corresponding to the two points and take their difference, Delta f(x):
Delta f(x) = f(x2) – f(x1).
Finally, to find m, you compute the ratio of the two Deltas:
M = delta f(x) / delta x.
When looking for m, we consider finite distances between any two given points, in the sense that the difference between x1 and x2 is finite. On the other hand, a derivative considers infinitesimally small distances between any two given points. Indeed, when writing down a derivative, we swap the symbol Delta with d, obtaining df(x)/dx. Considering infinitesimally small distances makes the derivative an extremely precise tool for understanding the behaviour of a function.
