The Missing Derivation Process

Old Yang was named Yang Shengguo, and he was already thirty-eight this year.

Watching Li Dong's chalk fly across the blackboard, he suddenly felt dazed.

In that instant, he seemed to see his younger self.

Back then, he had studied for both his bachelor's and master's degrees at Kyoto Normal University. Full of ambition, he had naturally planned to pursue a doctorate and embark on a research career.

But at the crucial moment before publication, his research findings had been stolen by the mentor he respected and credited under someone else's name.

He had felt wronged. Young and hot-blooded, he had not known how to bow his head, so he went to confront his mentor and caused a scene.

As a result, not only had his hopes of pursuing a doctorate been crushed, but he had also been quietly blacklisted within that circle. He could not even find a decent job.

His heart had already gone cold by then, so he returned to his hometown of Jiangcheng and became a math teacher at the ordinary public No. 7 High School.

During those first few years, he had wanted to teach those children well. He wanted to teach them knowledge, but even more, he wanted to teach them how to conduct themselves.

He wanted to teach them how not to suffer losses, and teach them that if they ever became teachers themselves, they must never take advantage of their students.

But reality was never as wonderful as imagined.

Every place had its own rules. The best students had all been snatched away by elite private schools and key high schools.

Most of the students who came to No. 7 High School were ordinary people of average aptitude. Some were still working hard but lacked the right methods, stuck in an awkward middle ground. Others had already given up completely, planning to coast their way to a high school diploma.

Watching them, Yang Shengguo's enthusiasm gradually cooled as well.

Just teach them. As long as he could live with his conscience and explain what he knew, whether the students listened or could learn it was up to their fate.

He had thought this was how his life would be—guarding his thermos of goji berry tea here as he slowly grew old.

Until today.

Li Dong, who normally could not even reach the passing line, had caught him off guard with a slap to the face—and given him a long-lost surprise.

At that moment, Li Dong saw nothing but the problem on the blackboard.

He was exceptionally focused now. Those complicated function graphs broke down before his eyes into their smallest units.

"No need to mechanically calculate where the derivative equals zero..."

"For problems with this kind of structure, the key is to observe the form."

"Since the problem requires the inequality to hold universally, we only need to find the critical point where the function graphs on both sides are tangent. Then the bound for the constant a will emerge."

He discovered that Joule's "decomposition method" was genuinely useful. As long as he saw through the core logic, all that remained was grunt work.

Though the grunt work was rather demanding.

He was still struggling a little. First, his concentration was only 0.1, so he was still easily distracted when facing intense mental calculations.

Second, his logical reasoning ability was still at the level of an ordinary person's 0. If he had not already solved a problem with the same underlying structure beforehand, he might not have been able to work it out so quickly.

Fortunately, his innate memory was 0.1, so he did not need to flip through textbooks for formulas.

Li Dong picked up the chalk. Facing the variation Old Yang had given him, he performed no complicated differentiation and did not draw any coordinate axes.

"Solution: The original expression is equivalent to e^x ≥ a ln(ex)."

Li Dong's hand paused slightly for half a second.

Then, he directly skipped the process of constructing functions, differentiating twice, and discussing extrema—a process that would have filled half a scratch notebook for an ordinary student.

Two of the most basic tangent-line inequality models appeared in his mind.

He wrote two extremely concise lines of derivation on the blackboard:

"By tangent scaling, we know: e^x ≥ ex and x ≥ ln x + 1."

"Therefore: e^x ≥ ex = e(x) ≥ e(ln x + 1) = e(ln x + ln e) = e ln(ex)."

When Li Dong wrote that chain of inequalities on the blackboard, the pen in Mi Xia's hand dropped onto her desk with a clatter.

"This is... double scaling?"

Mi Xia's face was filled with disbelief.

Old Yang was stunned as well. That leap revealed tremendous skill. Without an enormous accumulation of problems and extraordinarily sharp mathematical intuition, no one would dare write it like that.

Yet he could see that the logical chain within it was flawless.

"From the above inequality chain, equality holds if and only if x = 1."

"Therefore, a_max = e."

The chalk gave one final crisp scrape against the blackboard.

"Um... Teacher Yang, I'm done."

The classroom was so quiet that a pin dropping would have been audible.

The students looked at one another. They could not quite understand the answer on the blackboard, nor did they know whether it was right.

So they could only turn their eyes toward Old Yang.

Several students in the front row, including Mi Xia, were frantically checking the calculations in their scratchbooks.

"H-how did he get that step?"

A student muttered quietly.

"Where's the differentiation? He just used greater-than-or-equal-to signs directly?"

"Yeah, isn't that way too convenient?"

Old Yang stood where he was, staring at the solution on the blackboard without speaking for a long time.

After a while, he let out a deep sigh, as though he wanted to breathe out all the resentment that had been suppressed in his heart for so many years.

"Looks like... I can't keep muddling along and waiting to die anymore."

His voice was very soft. No one heard him.

Old Yang did not rush to judge whether the solution was right or wrong.

Because for the vast majority of Class 2, Year 3, this "Isomorphic Scaling Method" was a competition-level technique. There was no need to dwell on it.

Rather than waste time on that technique, it was better to make sure they secured the basic points.

"Li Dong, you..."

Old Yang had just started to say something when someone suddenly knocked on the classroom door.

Knock, knock, knock.

The door was pushed open, and a slightly weary face poked inside.

"Teacher Yang, sorry to interrupt. I need to say something."

Old Yang nodded toward the door.

"Go ahead, Teacher Jiang."

Teacher Jiang addressed the students below.

"Students, excuse the interruption."

"The city just sent out a notice. There'll be a physics competition next month, a selection event for the provincial preliminaries."

"The school has been assigned a few slots. Does anyone want to sign up? There's no training required—just treat it as a chance to get some practice."

The room fell silent, and everyone lowered their heads even further.

Teacher Jiang was used to it. Tasks like this, where they had to scrape together enough participants to meet a quota, were not exactly rare at No. 7 High School.

Students from their school mostly went just to make up the numbers. They were not really qualified to compete.

"If anyone wants to sign up, come find me in the office next Monday."

After saying that, he gave Old Yang a wry smile and a wink.

The meaning was obvious: He had no choice. He needed bodies.

Yang Shengguo nodded and watched Teacher Jiang leave.

After that brief interruption, the classroom returned to silence.

Yang Shengguo turned around and gave Li Dong a long look, his tone much gentler than before.

"All right, Li Dong. Go back to your seat first."

Li Dong felt his scalp prickle under Old Yang's stare. He always felt there was something strange about the way Old Yang was looking at him.

"Oh."

After Li Dong returned to his seat, Old Yang pointed to the blackboard and said,

"Li Dong's solution to this problem was... very correct."

"And his approach was extremely unconventional. He used the scaling method. It seems he really put in a great deal of effort in private."

Old Yang paused, then changed direction.

"However, this kind of solution places extremely high demands on logical thinking and is easy to get wrong. Everyone should judge according to their own ability."

"In the college entrance examination, problems like this are usually bonus questions. We should first secure the points we're supposed to get and not aim too high."

Although Old Yang was trying to cool things down, the way the students looked at Li Dong had already changed.

Surprise, suspicion, jealousy... all kinds of emotions were present.

"Didn't Li Dong fail math? How did he suddenly...?"

"Did he deliberately memorize this difficult problem just to show off?"

"Maybe. Otherwise, why didn't he even differentiate?"

In the front row, Mi Xia bit her lip tightly and continued deriving the solution in her scratchbook.

Even using the approach Li Dong had provided, she had only just verified the validity of the first inequality.

"Were the steps he omitted... really just too simple for him?"

Mi Xia glanced back at Li Dong, a faint sense of defeat rising in her heart.

The bell for the end of evening self-study finally rang.

It was Friday, and boarding students were allowed to leave school and go home.

Many parents were waiting for their children outside the school gates.

Li Dong walked toward the bus stop with his backpack on his back, while Liu Liang silently followed beside him.

It was not until they were about to board the bus that Liu Liang suddenly spoke. "Dongzi."

"Hm?" Li Dong stopped.

Liu Liang looked at him with a complicated expression and said in a tone that was half teasing and half serious,

"Have you really caught the study bug?"

Li Dong heard the trace of disappointment in his voice.

During adolescence, when a friend who used to fool around with you suddenly began working hard, that fear of being left behind was unavoidable.

Li Dong turned around and spoke sincerely.

"Liangzi, I think we need to take some responsibility for our futures."

"I want to give it a shot, even if it's just so my mom can worry a little less."

Liu Liang froze for a moment, then broke into a grin, hiding the loneliness in his eyes.

"Look at you, such high awareness. Go on, go on, future academic genius."

After saying goodbye to Liu Liang, Li Dong squeezed onto the No. 302 bus home.

It was already past ten at night, and the bus was nearly empty.

Li Dong sat by the window.

Outside, Jiangcheng's nightscape slid backward before his eyes.

The skyscrapers in the city center were brightly lit. The price of one square meter there was equivalent to his family's income for a month without spending a cent.

The bus slowly left the commercial district, and the lights gradually became sparse. The surrounding buildings turned low and shabby.

Li Dong tightly gripped the phone in his pocket. It held the key that could change his fate.

The bus stopped at the entrance to an old residential complex.

Li Dong climbed to the sixth floor with his backpack. Before he even opened the door, he heard his mother, Li Qin, talking on the phone inside.

"...All right, all right, then I'll have to trouble you."

"Oh, as long as his grades can improve. Besides, it's much cheaper than hiring a private tutor... Okay, then tomorrow. Thank you so much."

Li Dong pushed the door open and entered.

Li Qin hung up when she saw her son return.

"Xiao Dong, you're back? Are you hungry? Mom can heat up some food for you."

"I'm not hungry." Li Dong changed his shoes and looked at his mother.

"Mom, who were you talking to just now?"

Li Qin poured him some water and said with a smile,

"Oh, I wanted to tell you something."

"One of Mom's coworkers has a child studying at Jiangcheng University of Technology. He's a third-year student in the math department."

"I heard his grades are pretty good, so I asked him to come tutor you on weekends."

Afraid that her son might resist, Li Qin hurriedly added,

"He's a college student, so his rates are cheap. We can afford it."

Li Dong froze.

Jiangcheng University of Technology was a local second-tier university. And its math department... those who knew, knew.

She wanted a student from a second-tier university's math department to teach him?

Li Dong unconsciously touched his phone.

Who were his "teachers" now?

Isaac Newton, James Joule—some of the brightest stars in human history!

It felt like... he was eating an imperial feast in a group chat, only to come home and have his mother, worried that he was hungry, spend money to stuff two cold steamed buns into his hands.

He really wanted to refuse, but he saw the excitement on his mother's face.

He could not reject her kindness.

Ever since his father left, his mother had carried this family alone.

Now she was scrimping and saving to hire him a tutor. If he refused, she would think she had failed to provide him with the best learning conditions.

So Li Dong nodded and said,

"Okay."

"Don't worry, Mom. I really love studying now."

At those words, Li Qin's face immediately blossomed into a smile.

"Ah! That's more like it!"