What is a Derivative?

Every year I want to go back to a calculus book just to see how my ideas of limits, derivatives and integrals has changed this time, and I think its true for all kind of mathematics that if you revise them in different parts of life you will get different realizations and often epiphanies! throughout my life I have thought of calculus as hard math that Newton(Leibnitz) created to instantaneous change with time to best fitting slope on a curve to convergence of sequence, and I know its not end yet!

What derivative is? well, it’s function! but maybe slightly unusual one( very common if you’re into mathematics for a long time). This function takes an input(real number) and a function and returns an output(real number) which you can interpret as instantaneous speed of the function at that point. and by breaking down this processes we will learn some core concept of analysis!

     Derivative as function

Derivative is this “giant” function where you first set a function f and give it an input point x, what the derivative does is make an infinite sequence out of this one input point and whatever number/point this infinite sequence is converging to is the output point y. If you are unfamiliar with infinite sequence and convergence, Hold on for second!

(actually its more rigid than that, derivative technically check infinitely many type specific infinite sequences and if all of them converge to the same point that is count as the output)

let’s look at the traditional definition for a bit.

lim h → 0         f'(x) = {f(x + h) – f(x)}/h

its a definition for a hybrid sequence which depends on another sequence  <h>  which should be converging to 0, and there is possibly many sequence where <h> → 0, and no matter what <h> you choose this hybrid sequence f'(x)  should converge to the same output.

A schematic of a derivative

this definition has a very nice geometrical interpretation, actually the history is other way around, from this geometrical picture we originally got our definition!

Sequence of Geometric pictures

Here we have this sequence of triangular like shapes , and whatever this shape approach we take the height/base and exactly get the derivative at that point.


Convergence & Divergence

Now you can question why bother doing this definition with convergence of a weird sequence where the geometric picture is just fine enough, its because not all functions we can see and identify aha that’s the slope, this definition is workable even if we can’t visualize our function

But you can’t compute infinite things! how are we gonna say anything about them if we need to compute to the infinity! Well that’s where analysing comes in! this the reason I think this kind study is named analysis because its just nit about drawing lines or manipulate symbols
you have analyse the situation without getting into infinite computation

infinite sequence is the sequence of infinite things(very funny), but I think everyone has this innate feelings of what infinite sequence is so I am not going to formally define it here, rather I want take look at another gut feeling which is convergence and divergence.

the field Real analysis deals with convergence and divergence of sequence let me give you some example of diverging and converging sequence with out any context

Some infinite sequences that diverge
Some infinite sequences that converge

      World  of  new   math

This ideas of converging infinite sequences are so powerful that they opened up world for new mathematical character to come in which was rather impossible for Euclidean geometry and Khwarizmian Algebra to dream of.

The very concept of irrational numbers are defined to be set of all sequence of rational numbers which converge to them(so Pythagoras is kind of right all number might not be write using rational number but you can certainly approximate all of them with the rationals)

You can also construct weird sets like Cantors set as a limiting set

construct shape like Alexander horned sphere as limiting object of spheres

all the famous Mandelbrot set and the Julia sets are all defined on this kind approaching arguments.

As you can see this post less about calculus and more about a way of defining object which is very common in modern mathematics, I wanted to show like Calculus where derivative and integration are defined as limiting object all other fields have their own limiting objects and monsters with bizarre properties that defy all common sense ( maybe ill write next time about these)

I’m finishing this unorganized post by a concern of a Pioneer about these object, you decide yourself how to feel about them

“Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner.
In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that.
If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum.”
— Henri Poincaré, 1899

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