Leibniz believed that the derivative was the quotient of two infinitesimal quantities dy and dx, so he used dy/dx to represent the derivative. Although the derivative no longer means this, the set of symbols carefully invented by Leibniz is indeed very easy to use, so we continue to use it.
In other words, we still use dy/dx to represent the derivative today, but everyone must pay attention to the fact that dy/dx is a limit in the modern context and is no longer the quotient of two infinitesimal quantities.
If you are not familiar with the history of calculus, it is easy to have various misunderstandings about these symbols. This is also a major difficulty in many popular science texts and textbooks when teaching calculus. Because the idea of is new, but the symbols are old , it is indeed easy to get confused.
Therefore, in Leibniz, he first defined the differentials dx and dy representing infinitesimal quantities, and then used the quotient of the differential to define the derivative dy/dx, so the derivative at that time was also called the differential quotient.
But now the plot is completely reversed: we now define the limit using ε-δ, and then define the derivative dy/dx from the limit. There is no differentiation at all in , but due to historical reasons we still write the derivative as dy/dx.
So, do dx and dy, which were previously regarded as infinitesimal differentials, still have meaning now?
answer makes sense! The dx and dy of
are still meaningful. Of course, meaningfulness will definitely no longer mean the infinitesimal amount it used to mean. So, in the new context of ε-δ limit, what is the significance of dx and dy in the new era? Please look at the picture below:
The slope of the blue tangent line represents the derivative at point P. If we continue to use dy/dx to represent the derivative, then we can clearly see from the picture: dx represents the change in the x-axis, dy It just represents the change of the blue tangent line on the y-axis.
In other words, when the independent variable changes by Δx, Δy represents the change of the actual curve, and the differential dy represents the change on this tangent line. This is the meaning of the function differential dy in the new context. As for the differential dx of the independent variable, you can see that it is the same thing as the change in the x-axis Δx.
Because the tangent line is a straight line, and the slope of the straight line is certain. Therefore, if we assume that the slope of this tangent line is A, then there is such a linear relationship between dy and Δx: dy=A·Δx.
These conclusions can be easily seen from the figure, but whether a function is differential at a certain point is conditional. What we have here is a very "smooth" curve, so there is a differential dy at point P, which means that it is differentiable at point P. However, what if the function is a turning point at point P, a sharp inflection point? That won't work. Because if there is an inflection point, you can't make a tangent here at all, so why talk about Δy and dy?
Whether a function is differentiable at a point is a relatively complicated issue. The conditions for judging the differentiability of curves (univariate functions) and surfaces (bivariate functions) are also different. Intuitively, if they look "smooth", they are basically differentiable. The strict definition of
differential is as follows: Is there an infinitesimal A·Δx (A is a constant) that is linear with respect to Δy for Δy, so that the difference between it and Δy is an infinitesimal higher order than Δx. In other words, whether the following formula is true:
o(Δx) means the higher-order infinitesimal of Δx. Literally understood, the higher-order infinitesimal is infinitesimal than the infinitesimal. When Δx slowly approaches 0, o(Δx) can approach 0 faster than Δx. For example, when Δx is reduced to 1/10 of its original value, o(Δx) is reduced to 1/100, 1/1000 or even more of its original value.
If this formula is true, we say that the function y=f(x) is differentiable at this point, and dy=A·Δx is the differential of the function. Because this is a linear function, we say that the differential dy is the linear principal part of Δy.
This part of the content does seem a bit boring. The differential dy in Leibniz's time is an infinitesimal quantity close to 0 but not equal to 0, which is very intuitive to understand. However, the differential dy of the function we redefined through the limit of ε-δ turned out to be a linear principal part.This is very unintuitive, and the definition is quite difficult to pronounce, but this kind of calculus is modern calculus, a calculus with a solid foundation and strict logic.
In order to give everyone a more intuitive concept of this less intuitive differential concept, let's look at a very simple example.
We all know that the formula for the area of a circle with radius r is S=πr². If we increase the radius by Δr, then the area of the new circle should be written as π(r+Δr)². Then, the increased area ΔS should be equal to the difference between the areas of the two circles:
Did you see this formula? It is exactly the same as our Δy=A·Δx+o(Δx) above. It’s just that we replaced x and y with r and S. A here is 2πr. π(Δr)² here is the square term of Δr. Isn’t this the so-called higher order (square is 2nd order, Δr is 1 order, 2 is higher than 1) infinitesimal o(Δx)?
Therefore, its differential ds is the term 2πr·Δr:
Its geometric meaning is also very clear: this is the area of a rectangle with a length of 2πr (which is exactly the circumference of the circle) and a width of Δr, which seems to be The area of the resulting rectangle is obtained by "straightening" the circle.
Okay, that’s it for the differential thing. You can slowly experience the rest by yourself.
To be continued~
