Perhaps this will help some.

I seriously hope this will not offend you, but suspect it will.

Simply put, your post does not help me at all.

I am a programmer, not a mathematician, but given a formula, in a form I can understand(*), I am perfectly capable of implementing that formula in code. And perfectly capable of coding a few loops and print statements in order to investigate its properties.

What I have a problem with -- as evidently you do too -- is deriving those formula.

Like you (according to your own words above; there is nothing accusatory here), my knowledge of calculus is confined to the coursework I did at college some {mumble mumble} decades ago. Whilst I retain an understanding of the principles of integeration; and recall some of its uses, the details are shrouded in a cloud of disuse.

Use it or lose it, is a very current, and very applicable aphorism.

The direction my career has taken me means that I've had no more than a couple of occasions when calculus would have been useful. And on both those occasions, I succeeded in finding "a man that can", who could provide me with an understandable formula, and thus, I achieved my goal without having to relive the history of mathematics.

(*) A big part of the problem is that mathematicians not only have a nomenclature -- which is necessary -- the also have 'historical conventions' -- which are not; and the latter are the absolute bane of the lay-person's life in trying to understand the mathematician's output.

There you are, happily following along when reach a text that goes something like this:

We may think intuatively of the Riemann sum: Ʃ^b_a f(x) dx

as the infinite sum: f(x₀)dx + f(x₁)dx + ... + f(x_{H - 1})dx + f(x_H)(b - x_H)

Where did H come from? Where did a disappear to? Is H (by convention) == to b - a?

For the answer to this and other questions, tune in next week ..... to the last 400 (or sometimes 4000) years of the history of math