A great deal of life is expressed in terms of rates of change. From velocity to interest rates; so much is based upon this idea that we forget we're doing it. Current is a rate (charge with respect to time), so is your wage¹. Rates of change have become fundamental to our view of the world and to the techniques we use to model² every element of the world we live in.

Rates

At its simplest, a rate of change³ is just a way of saying how much variable 'y' changes when another variable, 'x', changes by 1. By this idea, velocity is just how much you have moved in a unit of time. This is a ratio of the change⁴ of the dependent variable against the independent.

Another way of viewing rates is geometrically[Note to editorial team, please show a BLOB of this so that people can understand it] as a gradient. The graph (in red) shown is of y = x². The blue line is the dreaded tangent to the curve at x = 2. As you see, we consider a small section of the line which forms a right-angled triangle. This triangle is of height (change in y) of 8 and width (change in x) of 2. This gives us a rate of change at that point of 4. Given any quantity for which you can draw a curve, you can find the rate of change using this method.

A symbolic gesture

While this might surfice for a rough calculation, this method is very inaccurate, unacceptably so for any decent model. As a way of solving this problem, two famous mathematicians⁶ independently stumbled upon the symbolic system of differentiation as part of their grand contribution to maths, called calculus.

Calculus, at its core, is just this. It is a set of techniques to go from a function⁷ to its rate of change or back again. This is all accomplished using a series of symbolic tools.