Ask anybody “what is a derivative?” and you’ll quickly find out that nobody can tell you exactly what it is in no uncertain terms without any question. If fact, most people you ask who have taken calculus can’t even come close. Think I’m wrong, try it and you’ll find out what I did, it took me eight years after college to find out and even then not exactly. That’s pathetic and unacceptable in my opinion. Of course I was programming the TI Calculators in calculus at the time so I had some interest to even ask the question. I mean, who cares right? No one out of college will ever calculate a derivative or integral again in any job outside of re-teaching it as a professor, so I/we understand that.
I had some interest because of programming step by step calculus into the calculators and while teaching tennis to this guy named Mike, I find out – he at one time worked at NASA. At the time of me teaching him tennis he had left NASA was a professional black jack player. Went all over the world making money at black jack in the casinos. I was also fooling around with on line poker at the time and asked him how come black jack and not poker? He said poker was too much gambling and black jack is relatively sure. He said he used differential equations to help him count cards and change the odds in his favor. Anyway I asked him what was a derivative? He said immediately that it was the slope of a tangent line to a curve. I said “but when I graph the derivative there is a line – but no tangent line and the slope is off.” He said the graph of the tangent line is meaningless, “of no value”. He said when you compute the derivative of a function at an “x” value you come up with a number and that is the slope of the tangent line at that point on the function. I thought even that was fabulous and an eye opener, but we had finished picking up tennis balls and so I let it go and started to again teach him tennis.
After that moment, I kept thinking and thinking and thinking of what he said and then it dawned on me. The number you get when computing the derivative at the chosen point “x”, no matter how deep the original function is the numerator of a fraction with the denominator = to 1. That was rise over run. If the numerator is 12, that is 12/1. You go 12 spaces up the “y” axis and one space over on the “x” axis. Draw a line from that point through the origin of a graph (0,0) and that is the exact angle or slope of the line of the derivative. For that line to me tangent, it must touch the curve at only one minute point, so that line has to be transposed to do that, however it will still be the exact slope. To transpose that line you have to compute the equation of that line to the desired point (point slope form) and then graph that function and you have the perfect picture of what a derivative is. That’s fabulous. (Equation of a line at a point on a curve is in my calc1 programs). Now at the next party you know all there is of “what’s a derivative” and can look like a nurd, I mean nerd. lol, Tom
p.s. (The exact angle of that slope by the way is tan^(-1)(12,1), Fabulous!!