Summer of Math Exposition

Presented by 3Blue1Brown 3blue1brown

DERIVATIVES – Intuitive Course

Audience:

Tags: derivatives

Hi everyone! This is my first video with math explanations, so I apologize in advance if anything was confusing or had slip-ups. I’ve tried to fix everything in the subtitles.

I’ve been passionate about math since childhood and was always a top student in school, though I was more about doing calculations and applying formulas. The real breakthrough came in 11th grade when I started preparing for the Unified State Exam (EGE) and got especially into parameter problems. I solved them on my own without peeking at solutions, and that significantly boosted my level. I scored 97 out of 100 on the EGE in 2025, though I still wasn’t completely satisfied – I’m something of a perfectionist.

After the EGE that summer, I kept studying math and started watching more English-language channels in the original, like 3Blue1Brown. This also improved my English – earlier it was hard to follow videos without subtitles, but now I often watch them without. I fell in love with intuitive explanations and realized how important it is to present material well. Once I got to university, I noticed how dry and fragmented math is taught there: they give formulas but don’t show the connections between them. I wanted to offer explanations that are intuitive, honest, and free from rote memorization.

At first, I started writing a detailed script for explaining the school curriculum, but then decided to focus on derivatives and make videos based on rough notes. Most of the work happened in June–July: I spent several days thinking through each idea and looking for the best explanation, cross-checked my hypotheses with AI, and came up with many explanations myself (e.g., for the derivative of power functions, composite functions, and the total differential). It was through tables that I figured out how to teach derivatives – and I haven’t seen that approach anywhere else.

During this time, I noticed especially how disconnected topics are in standard teaching: they give algorithms but don’t explain why they work or how they connect to what you’ve already learned. So in this video I tried to build a coherent picture of derivatives – not through examples, but through ideas.

I decided to join the Summer of Math Exposition (SoME) competition. For me, this is a chance to test my skills and show that understanding can be clear and vivid. I hope this video doesn’t go unnoticed. And yes – I finally published it!

Videos mentioned:

Timestamps:

  • 00:00 Introduction
  • 01:23 The derivative – instantaneous rate of change
  • 09:02 Derivative of e^x
  • 11:50 Derivative of sine and cosine
  • 14:47 Derivative of x^n
  • 20:43 Constants factor out of the derivative
  • 21:33 Derivative of a sum
  • 23:52 Derivative of a product (+generalization)
  • 27:10 Derivative of a quotient
  • 31:36 Derivative of a composite function (+a^x)
  • 35:16 Derivative of the inverse function
  • 36:11 Deriving the remaining standard derivatives
  • 41:38 Logarithmic derivative and bringing a function to base e before differentiating
  • 45:21 L’Hôpital’s rule
  • 50:06 Equation of the tangent line
  • 53:18 Higher-order derivatives and differentials
  • 57:46 Taylor’s formula
  • 1:06:41 Critical points (extrema)
  • 1:13:59 Functions of several variables. Partial derivatives and the total differential
  • 1:23:28 Tangent plane / hyperplane
  • 1:25:08 Directional derivative
  • 1:32:16 Derivative of an implicit function
  • 1:38:22 Gradient
  • 1:43:35 Higher-order differentials for functions of several variables
  • 1:47:36 Connection between higher-order differentials and binomial/polynomial coefficients
  • 1:57:06 Taylor’s formula in a nutshell
  • 1:58:39 Extrema of functions of several variables
  • 2:11:40 Constrained extrema
  • 2:25:28 Global extrema in a nutshell

Drawing app: DrawNote Subtitles made with riverside.fm

Analytics

3 Overall score*
121 Rank
5 Votes
5 Comments

Comments

3

The video has a good amount of thorough and substantive math, but the way it is presented made it difficult to follow and retain. The screen feels very busy, and the quick navigation thumbing around makes it difficult to know exactly where to focus. It is also a lot to put in one video. I would recommend splitting this up into several separate installments. Consequently, the video feels more like an extended walkthrough than an educational story. I would also recommend maybe trying to build the mathematics around a clear central question or insight rather than trying to cover so much.

4.1

I’ll be honest I did not watch the full video because it’s too hard for me. But I think your motivation is really good, and example is good as well. However, I still think some room to improve.

First, The pen you use to explain should be a different color otherwise its hard to see. Second, The English subtitle line change is kind of weird.

  1. Usually there should not be over 40 English characters in a single subtitle. Especially in your case, where you speak non English, so the only way we understand your words is by subtitle. A too-long subtitle makes
  2. There should not be any period in subtitle, because if there is a period, you should split a subtitle into two, therefore you don’t need speriodd

Third, you use infinity canvas and move around. but there is actually too many distracting things around the things you are talking about. I advice you either make those drawings into slide, or set them apart more in your canvas.

Forth, There is some meaningless pan and zoom that confuse the viewer when you are talking about things.

Finally, I think your audience should be at least undergraduate, it’s too hard for high school student that just learned calculus.

3.5

This video is in Russian without English translation which seems to violate the contest rules. Also the length of the video is a bit off putting as it’s 2 hours long which although the content is good, isn’t quite of the spirit of SoME.

1

Whizzing around a giant sheet of unstructured notes for two-and-a-half hours is useless.

1.5

I think it’s quite ambitious to include a whole course on derivatives here, which is definitely great for the math community to do. It is a bit long to review, especially for a math exposition, so it might be better to try to chunk the video into a playlists focusing on a few topics instead. Some of the text zoom in is a bit close and a bit fast, and can be slowed down slightly, as it makes the whole flow a bit difficult to follow.