Cell theory as alternative to Power formula and Chain rule #38

mixed results for new technique #39.1 Uni-text 8th ed.: TRUE CALCULUS without the phony limit concept

I am making a valiant effort to find a new technique of Calculus by using the complimentarity function to find either the derivative or integral of a given function. The results are mixed.

For the y = x^2 function, its integral is 1/3x^3 and its derivative y'= 2x and in the 10 Grid it goes beyond infinity at 3.162. So I am working in a rectangle that is 3.162 by 10. Now if I plug into the integral the number 3, I have an area of 9 and 3 plugged into derivative is slope 6. No apparent help there.

For the function y = x^3, its integral is 1/4x^4 and derivative is y' = 3x^2 and in 10 Grid it goes beyond infinity at 2.154, so I am working in a rectangle that is 2.154 by 10 for function and compliment function. Now if I plug into the integral the number 2 as close to 2.154, I have an area of 4 and a derivative of 12.
Now there is a curious aspect to this which was not evident in y=x^2. If we take the total area to be that of x^4 = 16 then the sum of the derivative of 12 with the integral area of 4 we end up with 16. Is that just a special random case or is that a pattern?

So I try it out on the function y=x^5 with integral 1/6x^6 and derivative y' = 5x^4. The number in which the function goes beyond 10 in 10 Grid is somewhere about 1.58 and the total area taking it to be x^6 is about 16 and the derivative plugging in 1.58 to that of 5x^4 is about 32.

So, I may have a pattern here, or, more likely, I have just glimpsed a closer look up front of how the integral is related to derivative in a cell.

And in this tinkering to find a new technique, I ask myself the question of whether beyond y = x^3 whether those higher exponents have any meaning in physics? Do we in physics ever run across the need of x^4 or higher exponents. I cannot remember the use of any higher exponents in physics other than those that daydream in physics. So it maybe the case that in mathematics, the Power formula is over used and that it really belongs to a narrow application of exponents 2 and 3 and occasionally 4.

I cannot think of a single physics application where exponent 5 is required..

AP