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.