Its a little sad to me that the ideas of calculus are only understood by those who make it all the way through long division, quadratic equations and trigonometry on their way to the final year of high school. There, they are taught a subject so simple that it could by understood by an 11 year old who doesn’t know her timetables (this overly specific example may be from my personal experience as a tutor).
I’m not going to explain how calculus was invented or how you can prove that it works, but I think the basic ideas of the derivative and integral are so relevant to any conception of how the world works that it is absurd that they aren’t taught to everyone by age 13.
How to get rich
Lets say Jimmy currently has wealth of $100 and it’s the first day of the year. If I start making $10/day, then we could say that the first derivative of my wealth would be my income which is $10/day. The numbers here aren’t necessary but might help people understand, the key idea here is that income is the thing that caused wealth to change, and hence we call income the first derivative. Why call it the first derivative? Because we can take more derivatives to getter a fuller picture of the situation.
If you want to get rich, you need to not only earn money, but earn it at an increasing rate. Let’s call the amount your income goes up by each year your yearly raise. Your yearly raise is what increases your income, and so we can call it the first derivate of your income. If you understood that, this would be a good time to stop reading and pat yourself on the back, because being able to identify that one quantity is the derivative of another is the core skill here, and what I think we should teach to every 13 year old (note that taking the first derivative is the same as taking the derivative).
Now that you have a solid understanding of calculus, lets move on to the second derivative. This is simple the derivative of a derivative. To go back to the old example, we had that your yearly raise is the derivative of your income which is the derivative of your wealth. Then we can say that your yearly raise is the second derivative of your wealth.
I’m a big believer that more examples = better, so these are some other examples of derivatives that you may find interesting.
- Speed is the first derivative of distance.
- Acceleration is the first derivative of speed and the second derivative of distance.
- Learning is the first derivative of understanding.
- The derivative of population is population growth (the fact that population growth is proportional to population means that this growth is exponential).
What exactly is the point?
While receiving the joy of patting yourself on the back because you understand calculus is great, it is isn’t the reason I think people should understand this. The reason you should understand calculus is because it is a powerful tool to think about the world. Let’s go back to Jimmy and his wealth.
If Jimmy got to choose between getting $100 now, or a $5 higher yearly raise, which one should he choose if he wants to have the most money in the long run? You don’t need to actually do any maths to get the answer, it is all about the relationship between the quantities. If one person has higher income than another, the difference in wealth will always be overcome eventually, and if one person is getting higher yearly raises than another, eventually the one getting higher yearly raises will end up with a higher income. Therefore he should take the higher yearly raise, this logic follows for all other systems where quanities are connected by derivatives, affecting the highest level derivatives has the greatest impact over the long term. It’s why people choose to go to uni rather than getting a job, you sacrifice a lower income now for the ability for your income to have a higher rate of growth in the future, and if given enough time the higher derivative always wins.
At some point in the future there will be a part 2 and maybe even a part 3 of this post, because I am trying to build a collection of plain english explainers of mathematical concepts that people learnt in awful ways. Any feedback on this one would be greatly appreciated.
