A Playful Foray into Derivatives
Author: Yi’an (Yet another essay submission)
Derivatives are a truly marvelous class of mathematical tools. Their beauty lies in the absence of the tedious, twisted, and inconceivable calculations of conic sections; instead, through the mere combination of a few simple, unadorned expressions—much like the Big Bang generating an entire vast universe from a single singularity—they birth an expansive realm of detail that not even the entire designated answer area can contain.
However, before I begin my exposition, I would first like to offer an alternative proof for Vieta’s formulas, employing a method akin to those used in solving many derivative problems.
This achieves the exact same effect as using Vieta’s formulas when the difference between the highest and lowest degrees happens to be 2, while in practice saving the time required to find a common denominator. Next is the proof regarding the product of the two roots in Vieta’s formulas; corresponding to the above, we use the form of adding the two equations together.
Next is a derivative problem that I personally find extremely typical, involving the addition and subtraction of expressions related to zeros, as well as the form , taken from the 2021 Guangzhou Comprehensive Test for Senior High School Graduating Classes (1)
First, we obtain two expressions
What is the reason for this? The reason lies in failing to immediately notice that x is one of the common factors. Even if complete factorization is impossible, we should always make sure to attempt factoring simple expressions when solving problems. At the same time, because the presence of lnx ensures that x>1 always holds true, dividing both sides of the equation by x keeps it valid, thus yielding the following
At this point, we seem to have once again fallen into an impasse. Yet, when we re-examine the prompt, we find that this discordant a does not make a second appearance in the problem statement. This perhaps suggests that we ought to utilize a as a bridge to our solution, for it truly serves no other purpose. Let us attempt to construct an equation in terms of a:
Presumably, after enduring the long trials and tribulations of the boundless sea of math problems, all of you seated here have already developed a soul-deep sensitivity to the expression on the right—that is, achieving our goal of substitution by constructing t as the quotient of x1 and x2. However, this requires the degrees of the numerator and denominator to be identical; for instance, in the expression below, the degrees of both top and bottom are 2
However, because the degree of x in lnx1x2 is 0, if we were to use the substitution method mentioned above, it would inevitably be incomplete.
At this juncture, we are required to emancipate our minds, seek truth from facts, and keep pace with the times—this is the living soul of Marxism. In other words, we must carry out a bold sublation of lnx1x2. Since, much like the a mentioned earlier, it offers no help whatsoever in solving the problem, why not try casting it aside:
At this moment, it is still difficult to establish the correctness of discarding lnx1x2. However, for someone with sensibilities as sharp as yours, did the phrasing of in the prompt not immediately remind you of that legendary chain of inequalities? Here, I shall write out its two-dimensional form from memory
Harmonic-Geometric-Arithmetic-Quadratic: the basic inequality comprises the middle two terms, while the content of the prompt closely resembles the fourth term, from which we can deduce
And how could such sensitivity be cemented by a mere few hundred practice problems? Let us return to the text above, letting t represent their quotient
Under the extreme tension of the examination hall, I dare say all of you might even recite the cos in the Law of Cosines as sin, or misread the abscissa as the ordinate, let alone handle this simple final step of finding a common denominator. My experience is to find the common denominator term by term before combining them, rather than drifting everything up in a single instant. I have had similar realizations with this little trick in conic sections, so allow me to interject a piece of content related to conic sections here
However, the most difficult part of conic sections is not setting up the simultaneous equations at all; quite the contrary, the absolute simplest part is combining the line and ellipse equations and finding the relationships between the various roots. It is only after this that “the road to Shu is harder than scaling the azure sky.”
For the expression above, my advice is to first calculate -2a2, then calculate km, and finally multiply them together, rather than foolishly calculating -2a2k right after -2a2, and only then getting -2a2km; the latter is prone to calculation errors.
Now, returning to the main topic: it was not until recently that it suddenly dawned on me why the problem bothers to tell us the magnitude relationship with a multiple between the two roots. The fundamental reason is that it wants us to divide them! This indirectly hints at using the substitution method to replace their quotient. Yet, dull-witted as your humble servant is, how could I possibly fathom the entirety of this problem? It is easy to see that
Usually, in cases like this where a zero seems to exist yet remains elusive, boldly plug in e,e/2,1,2,1/e for the variable. The test-makers have gone to great, painstaking lengths just to make the expression work out neatly to zero at that exact number. Thus
Let us look back for the nth time,
Heavy hangs the curtain of night, and the winter crows cry out in the cold; overcome by a creeping weariness, here I lay down my brush.
Posted on the night of May 14, 2022
Because I originally wrote this in google docs, I just randomly downloaded a plugin for the formulas, and it seems to have converted them into images. For convenience’s sake, I just brought them over in image format as well. I just looked up how to represent math formulas in hexo and markdown, and KaTex seems pretty good. I feel like I can post some more math stuff here in the future, and practice using KaTex to represent mathematical symbols while I’m at it. I feel it might also help improve my own problem-solving skills. After all, what I’ve covered above is such a minuscule fraction of the subject.