What does it mean to differentiate a function?

A function represents a quantity. For example the function s = (6t² + 4t) m, could represent displacement. The unknown t is inputted to find the displacement an object travels at a certain time.

The differencial of a function represents the rate of change of that function. So for displacement the differencial would be the rate of change of displacement. The rate of change of displacement tells you how how quickly the respective moving object is covering distance. This is velocity.

So the rate of change of displacement is velocity. When you differenciate displacement you get velocity. In the example s = (6t² + 4t) m , we know, by differenciating displacement, velocity, v = (12t + 4) ms^-1.

If you consider the velocity of an object. The rate of change of velocity is how quickly it increases or decreases. This is the accelleration of the moving object.

So the rate of change of velocity is accelleration. When you differenciate velocity you get accelleration. In the example we found v = (12t + 4) ms^-1, we know, by differenciating accelleration, a = 12 ms^-2

Summary:

DISPLACEMET >differenciate> VELOCITY >differenciate> ACCELLERATION

Note: The reverse of differenciation is intergration. Considering this, the following can be said:

ACCELLERATION >intergrate> VELOCITY >intergrate> DISPLACEMENT