What is the relationship between a function and its derivative?

The graph of a derivative of a function is actually the gradient of the original function. For example a straight line such as y=2x+3 has a constant gradient its derivative is a straight horizontal line, namely y=2.
For more complex function the gradient changes, for a parabola y=x^2 when x is below zero the gradient is negative thus the derivative will give a line below the x axis etc.

f = x^2  f' = 2x  f" = 2 (polynomial)

f = sin x f'=cos x f" = -sin x  (trig)

f= e^x f'= e^x  f" = e^x  

Graph them

f" == "Curvature" f' == "Bends"

The first derivative is the rate of change of the function. For instance, if the x-axis (horizontal axis) represents time, and the y-axis(vertical axis) represents distance traveled, then the first derivative represents the amount of distance traveled in an infinitely small time, i.e. the speed. The second derivative would represent the rate of change of speed per time, i.e. acceleration. The graph of the first derivative would be the graph of the speed vs time, and the second derivative would be the graph of acceleration vs time. It is best when first learning these concepts to relate them to physical examples whenever possible. However, at a very advanced level this is no longer possible, but that is far into your future. Anyway, good luck studying Calculus. I hope that you enjoy it as much as I did because it is really interesting.