value of the Chain Rule for both math and physics #40 Uni-text 8th

On Saturday, November 16, 2013 3:23:22 PM UTC-6, wrote:
So now, in the Chain Rule, what could be easier than finding the derivative of y = sin(2x) when using the Chain rule, so that we have the outer function of sin(u) and the inner function of 2x, so that the inner function derivative is 2 and the outer function derivative is cos(u) and the final answer of y' = 2cos(2x).



But we do not use the power formula to find the derivative of sine to be that of cosine, and that we learn that from other methods.



Well, I was looking for a alternative technique of finding the general formula of derivative and integral instead of relying so much on the Power Formula or the Chain Rule of its double power formula. So is there some alternative means of finding the generalized formula of derivative and integral? Perhaps not, and that a proof can be made that the Power Formula is the lowest means of a generalized formula. Because, taking the example of the identity function y = x where the derivative is 1 and the integral is 1/2x^2.. Is there an alternative method of area of a triangle that is not one half the area of the rectangle it is embedded within? Well, there is Heron's formula for area, by knowing the three sides, but such a alternative is so much more complicated than the Power formula.



What I am grappling with, is that the Cell theory is brand new to Calculus and revolutionizes calculus that it changes and alters much of Old Calculus, however, it seems to not do much for the Chain Rule and delivering the generalized formula of derivative and integral. Maybe that is the way it should be in that the Power Formula and Chain Rule are the reduced to lowest terms possible of fetching the generalized derivative and generalized integral.



Yet, there is the possibility I overlooked something.



AP

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Finally found it. It is the Fourier transform, which takes any function and decomposes it into both sine and cosine function. The Cell Theory of Calculus is a vast Fourier transform system.

Of course, it is not going to be easier than the Chain Rule with its Power Formula, but it does show how the Cell theory allows for the existence of the Fourier transform which can fetch the derivative and integral, since all we end up doing it switching from sine to cosine inside each cell.

Now the Fourier Analysis is far beyond the scope of college/university students and belongs in graduate school of mathematics. I shall only talk of it in Advanced Calculus.

AP