What does differentiation actually mean?

Differentiation is defined as the rate of change of a function. Imagine we had a function f(x), which we shall let equal x². We all know what the graph of x² looks like.

The derivative of x² is 2x. This means that, as the variable x increases, the gradient increases by 2x. In graphing a derivative, effectively the gradient is on the y axis, hence the steeper the curve of y, the higher the derivative.