What is a derivative?

Question

Mathematics

The derivative, by definition, is the limit of the ratio of the increase in the value of the function to the increase in the argument when the increase in the argument tends to zero.
I studied a lot of literature on this topic, tried to understand this derivative thoroughly, I know the definitions of the formula, I understand the geometric and physical meaning of the derivative.
But I do not understand one thing, how it turns out that the ratio of something can be equal. What does it mean in this case that the argument growth limit is zero? Does this mean that we accept its value as zero? (The expression "infinitely close to zero" I can't comprehend either).
Since the increase in the argument tends to zero, it turns out that the increase in the value of the function also tends to zero, and in fact it turns out 0/0 (uncertainty) (But what is this uncertainty, I don’t understand either? If anything, I know about the disclosure of certainty, but what does I don't understand the uncertainty.) But by calculating this very derivative, you can get a lot of different values ​​\u200b\u200b(both 3x, and 6, x2, etc.), but how does this work if the ratio is 0/0? According to my ideas, it turns out that dx = 0, dy = 0 and, in fact, there can be no relationship, because there are no changes in x and y. But as a result, the values ​​of the tangent of the angle of the tangent with the positive axis OX are somehow obtained.
Could you explain this in simple terms, even complex, but so that all this is understandable? Or is it necessary to somehow understand and represent it all in a different way? I don't understand :(

Answer 1

I studied a lot of literature on this topic, tried to understand this derivative thoroughly, I know the definitions of the formula, I understand the geometric and physical meaning of the derivative.

Do not fool yourself: if you understood the geometric meaning of the derivative, you would not have any further questions.
Let's start with an elementary geometric meaning: let's say you have a graph of a differentiable function y=f(x), it's such a continuous line. And you are very small and stand on this line. y points north, x points east, and the line goes northeast, for example. You are so small that it seems to you that the line near you is a straight line. The derivative f'(x) tells you how much the line is slanted to the east direction at x. Those. if f'(x) = 2 at x where you are standing, then if you go 1 step to the east, you will need to go 2 steps to the north to return to the line.
All these "infinitesimals" can be interpreted as "so small that the behavior of the function does not noticeably change", i.e. that the function is "almost straight" at this scale, and further reduction does not change anything.