89.35 Another look at the calculus power rules

TLDR: Combining two diagrams into a single diagram could provide a way of showing graphically the reason why these rules are connected, and I wonder, if it is possible to devise a mathematical formulation, based on this 'dynamic' geometry, that answers my naive question simply and convincingly.

Abstract: At point P on the graph of y = x\" (x > 0), a rectangle is formed by lines perpendicular to the coordinate axes and a right triangle is formed by a line tangent to the curve. The rectangle is divided by the curve into regions A and B, the triangle has base c. A previous article of mine [1] used two diagrams to support the algebra that showed how, on the one hand, AIB = n is obtained from the integral power rule and, on the other hand, x/c = n is obtained from the differential power rule. These were treated separately with no attempt to show a connection between them. This connection is, of course, a heavily-disguised version of the inverse relationship between the calculus power rules. This relationship is shown easily enough with a standard piece of algebra, but this begs an interesting, albeit naive, question: 'Why is an area rule the inverse of a gradient rule?' It occurred to me that combining these diagrams into a single diagram (see Figure 1) might provide a way of addressing that question, but I needed something other than a static diagram. I needed a dynamic picture, which I obtained by using the Java applet at [2]. The applet seems to suggest that, as P moves along the curve, the tangent line 'projects' the ratio AlB onto the x-axis. In other words, just as region B is always 1 / n th of region A, so the base of the triangle is always 1 / n th of the base of the rectangle. Since these ratios are simply 'boileddown' versions of the calculus power rules, this could be a way of showing graphically the reason why these rules are connected. I wonder, if it is possible to devise a mathematical formulation, based on this 'dynamic' geometry, that answers my naive question simply and convincingly?