Calculus (III): What Is A Derivative? How To Really Integrate Differentials?

You will need to master this to master calculus!

The moving car with ∫dx beside it — Illustrative art created by the author

When it comes to mastering calculus, a sound understanding of the notion of a derivative is of paramount importance. It is surprisingly easy to establish a functional understanding of the derivative without deeply questioning the fundamentals.

However, this approach is likely to cause critical problems when one applies it to advanced calculus/analysis. This essay aims to eliminate such a pitfall.

This just happens to be the third in the sequence of the most fundamental calculus concepts that I have covered so far in the calculus series. In my previous two essays in this series, I covered the notion of a function and that of a limit.

In this essay, I will start by discussing the ground fundamentals that make up a derivative. Then, I will extend the discussion to what it means to actually integrate in the context of a derivative. I hope you are as excited as I am to dive into these topics. Without any further ado, let us begin.

A Brief History of Time

Why don’t we begin by discussing the notion of time? We currently stack up 60 minutes into an hour, 24 hours into a day, and 7 days into a week…