﻿In this text I will explain:
* What is the derivative
* The story of what the derivative is for multidimensional functions


Given a function f(x), we can talk about stating the limit of the function at a point being equal to a value L.


A limit is essentially just a restriction of some sort about the function. A property almost. In the same way you might say “this function has height 2 when the x=1”, you can also say “the limit of f(x) when x=1 is 2”.


This means limits are interpreted as nothing but properties.


Now the way that the derivative is defined is sneaky. The derivative is a function f’(x), and it is defined internally in terms of tuples (x, f’(x)) for all x in R. Now the way that f’(x) for a specific x is defined is in terms of a composite function, which secretly stands for the slope at that point.


The composite function is: (f(x+h) - f(x))/h


And we take the limit of this composite function as h->0


What does “take the limit” in this case? It means find a valid value for L such that the limit is true for that function at that point. If you can’t find a value L, then we choose to have that point not be defined.


And so that’s how the derivative function f’(x) is defined.


Now in the multidimensional case it’s more tricky. We want to ask the same question about finding the slope value function, but the problem is that we don’t know which direction to take. One thing we can do is give the direction as an argument of the function. This gives us the directional derivative. But this isn’t a good derivative for multivariable functions, because it misses a few of the properties we expect of derivatives. In fact the correct definition of derivative is in terms of a matrix B. And you ask the same question you did with L, except you ask it with B. In a sense, it’s like if the directional derivative fails because it only picks one direction, but using a matrix you can say “I pick all the directions”.


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In Newton spacetime, we have Lagrange, who asked a very interesting question: can Newton's second law be encoded into the surrounding spacetime, such that objects without a force take an “antiparallel curve path” on the surface of the space.


This turns out to be false because using the Poisson gravity + Newton’s laws, you fail to get an equation that can conform to the auto parallel curve equation. But if you include time with space, then everything works out and yes you get an auto parellel.