The "first" Monkey Rule is really the 0th Monkey Rule. You probably aren't going to be surprised by it at all. [br][br]To lead you towards it, I ask you: What is the slope of [i]all of the tangent lines[/i] of the linear function [code]f(x)=mx+b[/code] below? [br][br]Feel free to adjust m and b, and also to move [code]A[/code] along [code]f(x)[/code] to make conjectures and explore. As in earlier activities, point [code]B[/code] was generated with the code [code](x(A),slope(g))[/code] and is tracing the derivative of [code]f(x)[/code].

As we see above, the slope of the tangent line of a linear function is always just the slope of the linear function. The reason for this is simple: because linear functions are lines to begin with, their tangent lines are just themselves. [br][br]Since the slope of the tangent line to a linear function is just the slope of the linear function, we can deduce the 0th Monkey Rule[br][br][b]Monkey Rule 0[/b]: The derivative of a linear function[code] f(x)=mx+b [/code]is constant, and equal to the slope of the linear function. In other words [code]f'(x)=m[/code].[br][br]We can see Monkey Rule 0 in action above by noticing that as we move point [code]A[/code], we see that point [code]B[/code] (which traces the derivative of [code]f(x)[/code]) is always at height m above the x axis. [br][br]In the next activities we'll get into some more intricate Monkey Rules. Some people actually find them easier than this one since there's a bit more to latch onto. Onward![br]