Calculus

Derivatives

Derivates are an expression of the instantaneous change with respect to time. so they are expressed as dx/dt. Because they are attempting to measure the change at a given instant or moment, by definition they are already problematic, because change by definition does not occur at an instant, rather it is a measure of change over time, and hence the whole concept of “infinitesimal” times are instriduced, which is really an imaginary quantity that seeks to bridge this fundamental contradiction in our concept of change.

Plotting change for simple functions against time on the x axis enables us to see that the instantaneous change at any point on the resulting curve is given by the slope of the curve at that point. This is quite intuitive- a more steeply sloping change, the kind that gets everyone excited when watching the poll results, house prices or financial markets is indicative of a more rapid growth, whereas flatlining is indicative of a lethargic or dead market, which is good for personal weight records, cholesterol levels etc., you get the point. The slope at any point, in turn is the relationship between the two sides of the infinitesimally smal triangle at that point whose sides are given by dx and dt, and whose hypotenuse is itself the slope. So for example, where there is a high ratio between the height and breadth of the stairs, you would get a steeply sloping staircase.

The practical applications of this observation is measurements of quantities that involve such infinitesimal changes, like the area of a circle. The reason that the circle is such a beguilng figure is because it is a collection of dimensionless points which bend around each other by an immeasurable amount- if it were measurable, then the points would not be dimensionless. But at the same time we can have an idea of the overall result of such infinitesimal changes, which is a circle. But because our reprsentation of points and a circle are approximations, so also are our measurements of such objects, and this is where derivatives come in.

The Area of a Circle

A circle, as we have been saying is an example of necessary approximations in our measurements of objects. We think of quantities such as the length of a line or the area of a tetrahedron as definite quantities, although even here, we ignore infinitisemal quantities like the inherent curvature of surfaces on which that measurement is occuring. In order to specify the are of a circle, one should then be able to insert a definite amount of squares, or tetrahdrons into that circle, since after all, that’s how we express area, isn’t it?- as”square” metres. We’ve defined area as an approximation of the number of squares we can get into it.

Here lies the problem- no matter how tiny the squares near the circles edge, there will always be an area between the edges that remains unaccounted for, because the points being infinitesimally small and dimensionless, we can always “zoom in” an increase the resolution to expose the irregularities that might have become “smoothed over” in our eyesight. It’s more or less precisely the problem of fitting a square into a circular peg, it turns out we can do it- approximately. It turns out also that babies are not silly to try and have a go at it. Approximations are highly scientific.

So this is how derivatives deal with the problem: Rather than squares, imagine a set of rectangles wrapped around each other like rubber bands to obtain the circle, or like a child’s wooden toy with concentric rings, you know, all put together a central pegto make a big circle. Each of those bands has a length and height, being rectangular, and all the heights put together equal the radius of the circle. Each subsequent band is taller than the first, of course, like the child’s toy, so you can arrange them all neatly in a row along the x-axis, can’t you? That’s your area of the circle- its the area of that triangle. The tallest band on the end is one side, its length is 2piR, the circumference, while the other side is simply all the heights added together, which as we have said is the radius, R. The area of that triangle is 1/2 times 2piRsquared. which is piRsquared. Voila, dudes, easy. But that’s still an approximation, as we have seen.

So where is the approximation represented in the formula? I suspect it is in the nature pi, but I’m not sure. Well, it must b, because we never explained how pi got there in the first place. pi of course is the ratio of the circumference and radius of any circle. That we never manage to stop at a partucular decimal is reflective of the fact that as we increase the resolution to orders of ten, the uncertainty remains, infinitely.

Euler’s number and the derivative of exponents

An exponential function is essentially any number raised to an increasing power. When plotted this is a curve of increasing slope. The bigger the base number, the steeper the slope, because the rate of change depends on the number. If the number is 2, quantity is only doubled each time its multiplied by 2. But if it is 3, each time its tripled and if it’s say, 33, each time it’s mutiplied by itself, it’s “33’ed”. So if you had one farm, you got 33 farms. The next time you go crazy, you now got (wait a second while I do the um..mental math) 1089 farms. Yes, you’re just sat there with 1089 farms. You can’t even get around them in a day. The next time you got 35,937. In just three steps, now you can’t even count them in a day (just out of curiosity, the next steps are 1,185,921, 39,134,393, 1,291,424,969- ChatGPT took full 3 seconds to get that last one).

So an exponential always starts at “1” (anything to the power of 0), but the first step is the base number itself (raised to 1 is the number itself). The next step is as many copies of the base as the base itself. So the bigger the base, the more copies of it. You’ve added the number to itself by as many times as the number itself. So if you started with 2, you only get another set of 2. If 2, you now have three sets of three. That’s not a cube, its a square. if 33, then you now have 33 33’s, like the amazed farmer. Every time, you get a number of copies of whatever you had in the previous step equal to the base number- THAT’s the principle of the exponent. The increase at any particular point depends only on the base number.

So to give an example, if you got to a million through a base of 2 as well as through a base of 33 (albeit through differing amounts of steps), the instantaneous change from there on, only depends on the base (granted that this is a fictitious example): in the case of base 2 the next step is 2 million, in the case of base 33, you now have 33million. If you noticed, I was able to calculate the change even (albeit fictitiously) without even going through the initial calculations. The difference in increase was 31million (the first increased by 1million and the second by 32 million).

So you see that the increase is not “equal to” the base, rather it is merely proportional to it, that it, it depends on it in some respect, it is some multiplication of it, at any point on the graph. Now because this “increase” is a time dependent function, we get a specific increase if we introduce a specific time along the x-axis, but because we now fix this time, we get a fixed increase and thereby we fix the proportionality constants to each base. So for the base 2, this value is , for 3 it is , for 4 it is …. and so on. We are tempted to see an exact reflection of the number in the increase in the number, but this is not the case, because of the effect of time.

There is a number for which it is 1 and this number is e which is 2.71818, “Euler’s number”, in the case of which it is an exact reflection of the number itself in the infinitesimal time.