A Bit More About Monkey Rule 2
Before we move on to the other Monkey Rules, let's take a second to talk about Monkey Rule 2. Plotted below is the constant function f0(x)=3 and its derivative f0'(x)=0. I've also added the point A to the graph, and a tangent line to f0 at A. The tangent line always has slope 0, and so you can barely see it since it coincides with the graph of f0. This is the geometrical reason why the derivative of constant functions is 0. Notice, if you change 3 in the definition of f0 to 4 or another constant, nothing changes. This is why Monkey Rule 2 applies to all constant functions. Try it out! Double click on f0 and change 3 to another constant. If you set it outside -1 to 4 you won't see it on the graph without adjusting the window.