This applet is designed to help you understand what happens when you differentiate linear, quadratic and cubic graphs.
The general cubic has the form: y = Ax³ + Bx² + Cx + D.
The general quadratic is: y = Bx² + Cx + D.
The general linear is: y = Cx + D.
These curves are drawn in white and their equations given at the top left. Varying A, B, C and D changes the curves. (Note that if you're in quadratic mode, A no longer is relevant, so clicking it will have no effect. Similarly, in linear mode A and B are not used.)

The differential graph (gradient graph) is drawn in blue. Play around until you can see and understand the relationship between the two graphs. This will deepen your understanding of differentiation.
Challenge - Pressing "Random Curve" will generate a curve (or straight line) in yellow. Try to cover the yellow curve with the blue one by moving the white one!(It sounds more complex than it is!) You'll have to think about it!
Enjoy playing around with this very profound area of mathematics!