Derivative Calculation: Definition, Rules and Techniques | 导数的计算：定义、法则与技巧

📚 Derivative Calculation: Definition, Rules and Techniques | 导数的计算：定义、法则与技巧

In A-Level mathematics, the derivative is one of the most powerful tools in algebra and analysis. It captures the idea of instantaneous change and enables us to solve problems in geometry, physics, economics and beyond. This article reviews the definition of the derivative, the key rules for differentiation, and the techniques needed to differentiate complex functions confidently.

在 A-Level 数学中，导数是代数和分析中最强大的工具之一。它刻画了瞬时变化的概念，使我们能够解决几何、物理、经济等领域的问题。本文回顾导数的定义、微分的关键法则，以及自信地求导复杂函数所需的技巧。

1. Definition from First Principles | 从定义出发的导数

The derivative of a function f at a point x, written f′(x), is the instantaneous rate of change. Using the idea of a limiting average rate of change, we obtain:

函数 f 在点 x 处的导数，记作 f′(x)，是瞬时变化率。利用平均变化率的极限，我们得到：

f′(x) = limₕ→₀ [f(x+h) − f(x)] / h

This limit is often called differentiation from first principles. If it exists, we say f is differentiable at x. Geometrically, f′(a) gives the slope of the tangent line to the curve y = f(x) at x = a.

这个极限常被称为“从定义出发求导”。若极限存在，则称 f 在 x 处可导。从几何上看，f′(a) 给出曲线 y = f(x) 在 x = a 处切线的斜率。

Example: For f(x) = x², the first-principles calculation gives:

例：对 f(x) = x²，从定义出发计算得到：

f′(x) = limₕ→₀ [(x+h)² − x²] / h = limₕ→₀ (2x + h) = 2x

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2. Notation | 导数的记号

There are several standard notations for derivatives. If y = f(x), the derivative can be denoted by f′(x), dy/dx, or Df(x). In this article we use both Lagrange notation and Leibniz notation.

导数有几种标准记号。若 y = f(x)，导数可记作 f′(x)、dy/dx 或 Df(x)。本文将同时使用拉格朗日记号和莱布尼茨记号。

Notation | 记号	Name | 名称	Example | 示例
f′(x)	Lagrange | 拉格朗日	f′(x) = 2x
dy/dx	Leibniz | 莱布尼茨	dy/dx = 2x

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3. Basic Rules of Differentiation | 基本求导法则

The simplest rules allow us to differentiate power functions, constants, constant multiples, sums and differences.

最基本的法则使我们能够对幂函数、常数、常数倍以及函数的和与差求导。

Rule | 法则	Function | 函数	Derivative | 导数
Constant | 常数	c	0
Power | 幂	xⁿ	n xⁿ⁻¹
Constant Multiple | 常数倍	c·f(x)	c·f′(x)
Sum/Difference | 和/差	f(x) ± g(x)	f′(x) ± g′(x)

The power rule is valid for any real constant n. For example, if y = x⁵, then y′ = 5x⁴; if y = √x = x^1/2, then y′ = ½ x^−1/2.

幂法则对任意实常数 n 都成立。例如，若 y = x⁵，则 y′ = 5x⁴；若 y = √x = x^1/2，则 y′ = ½ x^−1/2。

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4. Product Rule | 乘积法则

If u(x) and v(x) are differentiable, then the derivative of their product is u′v + uv′.

若 u(x) 和 v(x) 可导，则它们乘积的导数为 u′v + uv′。

d/dx [u(x)v(x)] = u′(x)v(x) + u(x)v′(x)

Example: Differentiate y = x² sin x. Let u = x² and v = sin x. Then u′ = 2x and v′ = cos x, so

例：对 y = x² sin x 求导。令 u = x²，v = sin x，则 u′ = 2x，v′ = cos x，所以

dy/dx = 2x sin x + x² cos x

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5. Quotient Rule | 商法则

For a quotient u(x)/v(x), provided v(x) ≠ 0, the derivative is (u′v − uv′)/v².

对于商 u(x)/v(x)，当 v(x) ≠ 0 时，其导数为 (u′v − uv′)/v²。

d/dx [u(x)/v(x)] = (u′(x)v(x) − u(x)v′(x)) / [v(x)]²

Example: y = tan x = sin x / cos x. Let u = sin x and v = cos x. Then u′ = cos x and v′ = −sin x. Thus

例：y = tan x = sin x / cos x。令 u = sin x，v = cos x，则 u′ = cos x，v′ = −sin x。因此

dy/dx = (cos²x + sin²x) / cos²x = 1 / cos²x = sec²x

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6. Chain Rule | 链式法则

The chain rule differentiates composite functions. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.

链式法则用于

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