Product Rule for Derivatives
Product Rule for Derivatives
This is a visualization of the product rule in calculus, which says that, for differentiable functions and ,
Or, equvalently,
The basic idea of this visualization is that any product can be visualized as the area of an rectangle. In this case, the rectangles will be rectangles for different values of .
For small , can be approximated as . We can visualize the numerator as a change in rectangle area from an rectangle to an rectangle. That change can be visualized as the L-shaped area made of three rectangles colored red, brown, and blue in the activity above. If we define and , then it's clear that the area of the red rectangle is , the area of the blue rectangle is , and the area of the brown rectangle is .
Now drag the slider to the left, making is smaller. What you can see is that, as gets smaller, the brown rectangle becomes less and less significant in the L-shaped area representing the difference in rectangle areas. Ignoring that brown area, we can see that
.
In the limiting case as ,
One slightly confusing thing about the picture is that the graph shows a parametric curve where varies invisibly while the curve represents values of . Don't let that confusion get in the way of the visualization.