Frenetic Array

A canvas for logic and imagination.


As I go throughout my undergraduate studies, I have gotten really good at digging deeper until I completely understand a subject; however, when I observed around me, I realized that I am in the minority. Often times, students only desired to know how to get from question to answer without any intermediate thinking. When trying to hypothesize an answer, I cannot think of a definitive one. However, I know one possible contribution: encapsulation1

The most famous example I came across is the derivative of any trigonometric function besides $\sin x$ or $\cos x$. As any student who’s taken Calculus can tell you, $$\frac{d}{dx} (\tan x) = \sec^2 x $$

However, most students couldn't tell you:

$$\begin{eqnarray} \frac{d}{dx} (\tan x) & = & \frac{d}{dx} (\frac{\sin x}{\cos x}) \ & = & \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \ & = & \frac{1}{\cos^2 x} \ & = & \sec^2 \end{eqnarray}$$

At first glance, it's hardly useful information. As a matter of fact you might think of it as piece of trivia — until you see the amount of people that miss it on a quiz or test. And entry-level Calculus is not the only place this is relevant; physics, thermodynamics, mathematics of any kind, computer science, and the like.

A question you might be thinking, why does it matter? If it's worth a couple of points on a test, is it that harmful? Maybe not, until you realize you conceptually do not understand the subject — only mechanically.

The next time you're learning something, ask yourself: "why?". The outcome might delight you.

  1. The act of enclosing in a capsule; the growth of a membrane around (any part) so as to enclose it in a capsule. ↩