📚 Deriving Derivative Formulas from First Principles | 从第一原理推导导数公式
The derivative is arguably the most important concept in calculus. While many students memorise differentiation rules, understanding where these rules come from — using the formal definition of the derivative — is essential for IB Mathematics Higher Level. This article walks you through the first-principles derivation of the most common derivative formulas, step by step.
导数可以说是微积分中最重要的概念。虽然许多学生通过死记硬背来掌握微分法则，但理解这些法则的来源——即利用导数的形式化定义——对于 IB 数学高级水平课程至关重要。本文将逐步带您通过第一原理推导最常见的导数公式。
1. The Definition of the Derivative | 导数的定义
The derivative of a function f at a point x is defined as the limit of the average rate of change over an interval [x, x+h] as h approaches zero:
f ′(x) = lim(h→0) [f(x+h) − f(x)] / h
This is known as differentiation from first principles. Geometrically, it represents the slope of the tangent line to the curve y = f(x) at the point (x, f(x)).
函数 f 在点 x 处的导数定义为当 h 趋近于零时，区间 [x, x+h] 上平均变化率的极限：
f ′(x) = lim(h→0) [f(x+h) − f(x)] / h
这被称为"从第一原理求导"。从几何角度看，它表示曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。
Before we begin, let us recall three standard limits that will appear repeatedly. They are usually proved using geometric arguments or the squeeze theorem:
lim(θ→0) sin θ / θ = 1, lim(θ→0) (cos θ − 1) / θ = 0, lim(h→0) (eʰ − 1) / h = 1
在开始之前，让我们回顾三个会反复出现的标准极限。它们通常通过几何论证或夹逼定理来证明：
