This is a visualization of the product rule in calculus for functions that are product of two functions.
For example, . This is a product of and ;
We are going to look at a function . The basic idea is that any product of two functions can be visualized as the area of rectangle. Please note that the curve below is parametric, meaning that each point on the curve represents (u(t), v(t)) for some t, not the (t, f(t)).
Steps and questions to consider:

Set t=1. h=0. Let's say this is our "initial" rectangle.

Change h to 1 slowly. What's happening to the rectangle as you increase h from 0 to 1?

Write an equation to describe the change in the area of rectangle. (Final - initial)

Slowly change h to approach 0. What do you notice about the rectangles?(i.e. at h=0.02, do you still see the pink rectangle? Why or why not?)

Write an equation to describe the very small change in the area of rectangle.

Questions to answer
Remember that we just looked at the change in area. In other words, change in y or .

Using first principles, write the derivative function of . In other words, .