what is the geometrical meaning of the derivative of a function f?

The derivative of a function f(x), usually denoted as f'(x), gives you the gradient of the graph of f(x) for every point x in the domain of f. Therefoore, the derivative of a funtion can show how fast/slow a function increases/decreases/fluctuates. So, for example, if you consider f(x)= 2x+3 , its derivative f'(x) is equal to 2 (f'(x)=2). This means that the gradient of the straight line f(x) is the same for all x and that it is increasing, since f'(x)=2>0.