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Linkage between finite differences and calculus.

click for basic natural numbers from sets and set membership.
Click on blackboards for better resolution.

Think about why the derivative of x^2 is 2x (look at binomial expansion of (x+a)^2 ) "Let epsilon > 0 be given" is a way around a vanishing but necessary variable (in this case "dx") be manipulated.

The function name df() is not special in any way I just chose it for the happy coincidence at the end, I could have called it mashedpotatoes(). The variable name dx is not special in any way I just chose it for the happy coincidence at the end, I could have called it q or fred.
So it is no accident that the derivative of x^n is n*x^(n-1) this falls out directly from the 2nd term binomial expansion of the (x+dx)^n
The first term is simply x^n and when the subtraction of the original x^n happens in the finite difference the first term goes away.
The second term is n*x^(n-1)*dx all other terms have dx's of higher powers. During the division of dx on both sides, this second term is the only one without any dx terms [ n*x^n-1)*dx/dx ] all the other terms have higher terms of dx, and, in the limit as dx (epsilon) goes to zero these all dissapear.

Derivative of a sum is simply the sum of the derivatives.

I'm using the Taylor series for Sin and Cos to find the important bits that predominate as dx epsilon goes to zero. Again I'm leaving out steps for clarity and space.

Also don't let it throw you that derivatives are sometimes shown as functions or functions that work on functions. You can always think of the derivative of a function as another random function that just happens to have the properties of showing the rate of change of that first function. And one can always use the finite difference trick to find the function that happens to be the derivative (although its probably only necessary until you get the connection between finite differences and the mechanical operation of derivatives.)

Derivation of Natural numbers, Integers, and Rationals, from Set and Set Membership.

Note: the impetus for making set represenations of numbers contain all 'lower' numbers is to impose an order on those numbers. And in fact is used later on when subsets are used for 'less than'. The impetus for using the empty set as 0 was simply the empty set is the simplest set possible.

So that last little bit that ties in arbitrary numbers of increments shows how to add any Natural numbers.