This applet aims to explain the geometric meaning of the derivative.

Move the point A using the mouse: Δx will change. How does the ratio of the increments change?
Then show the tangent at P and compare it with the secant PA when A moves on the curve y=f(x). What can you observe?
Now repeat the previous steps, but changing the value of x_0 (drag it with the mouse along the x axis).
What does it change when P is on the left or on the right of the maximum? Why?
What does it happen if P is exactly on the maximum? Why?