Differentiation is a useful tool for determining how fast something is changing, which is represented by the rate of change of a function or the slope of a graph. It's key when you're looking at movements, shapes, or physics simulations. That rate of change, or a function that stands for it, is what we call a "derivative". The process of finding the derivative is called "differentiation".


<aside> 💡 英語だとそれぞれを ”derivative”、”differentation”と別の単語で表すので気をつけてください。また日本語ではある関数を微分した結果得られる関数を「導関数」と呼ぶこともありますが、英語には直接対応する単語は無いようです。


This page has become a bit lengthy, but the main goal here is to show that the basic idea of differentiation is actually quite simple. At first glance, formulas involving derivatives can seem cryptic. However, especially when taking a numerical approach that we’ll look at, it's just a matter of finding the difference between neighboring points and dividing that difference by the distance.


<aside> 💡 I'm not saying that it's trivial or that I fully understand the subject. In fact, it's quite the opposite. I have only scratched the surface. But differentiation can be very powerful and useful just by understanding the basics and learning a few simple practical methods. 微分が取るに足らないものだったり、既に完全に理解したと言っているわけではありません。実際はその逆で、ほんの表面をかいつまんでいるだけなのですが、それでも基本を理解し、いくつかのシンプルな実用的な方法を学ぶだけで、微分は非常に強力で役立つものになります。


Rate of Change



Think of an apple falling straight downward. From the starting point, the apple's position changes moment by moment, moving downward. The rate of change of the apple's position with respect to time is its velocity. Velocity can be defined as the derivative of position with respect to time. As the apple falls, not only does its position change, but its velocity also increases due to gravity. The derivative of velocity with respect to time is called acceleration, which represents how much the velocity changes at any given moment. In this example, the gravity acting on the apple is constant, so the acceleration also remains constant.


| Change of position | → Differentiate → | The rate of change of position =Velocity =The slope of the graph of position | | --- | --- | --- | | Change of velocity | → Differentiate → | The rate of change of velocity=Acceleration =The slope of the graph of velocity |

| 位置の変化 | → 微分 → | 位置の変化の割合=速度 =位置のグラフの傾き | | --- | --- | --- | | 速度の変化 | → 微分 → | 速度の変化の割合=加速度 =速度のグラフの傾き |

The rate of change can be measured not only over time, but also over space. For example, the slope of a hill indicates the rate at which height is changing relative to the horizontal position. Therefore, it is the derivative of height. If we know the derivative of temperature relative to position, we may be able to predict the flow of wind.