Differentiation Core Rules and Applications for Edexcel A Level | 爱德思 A Level 数学微分核心法则与应用

📚 Differentiation Core Rules and Applications for Edexcel A Level | 爱德思 A Level 数学微分核心法则与应用

Differentiation is one of the central pillars of Edexcel A Level Mathematics. It provides a way to measure instantaneous rates of change, analyse the shape of graphs, and solve optimisation problems in mechanics, economics, and geometry. This guide covers the core rules, standard results, and common exam-style applications you need to master.

微分是爱德思 A Level 数学的核心支柱之一。它提供了度量瞬时变化率、分析图形形状以及求解力学、经济学和几何中最优化问题的方法。本指南涵盖你需要掌握的核心法则、标准结论和常见考试应用。

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1. The Idea of a Derivative | 导数的概念

A derivative measures how a function f(x) changes as x changes. Geometrically, f'(x) gives the gradient of the tangent to the curve y = f(x) at a point. For a curve, the tangent gradient varies from point to point, so the derivative is itself a function.

导数衡量函数 f(x) 随 x 变化的快慢。从几何上看，f'(x) 给出曲线 y = f(x) 在某一点处切线的斜率。对于曲线，切线斜率逐点变化，因此导数本身也是一个函数。

When you are asked to ‘differentiate’ a function, you are being asked to find its derivative. In Edexcel exam papers, derivatives are written as f'(x), dy/dx, or d/dx [f(x)].

当你被要求对函数“求导”时，就是要找到它的导数。在爱德思考试卷中，导数通常写作 f'(x)、dy/dx 或 d/dx [f(x)]。

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2. Differentiation from First Principles | 从第一性原理求导

The formal definition of the derivative is based on the gradient of a chord between two points that get infinitely close. The first principles formula is:

导数的正式定义基于两点无限接近时弦的斜率。第一性原理公式为：

f'(x) = lim (h → 0) [f(x + h) − f(x)] / h

This expression represents the limiting value of the average rate of change over an interval of width h. In the exam, you may be asked to differentiate a simple quadratic or reciprocal function from first principles.

该表达式表示宽度为 h 的区间上平均变化率的极限值。在考试中，你可能需要从第一性原理求一个简单二次函数或倒数函数的导数。

For example, if f(x) = x², expanding f(x + h) = x² + 2xh + h² gives [2xh + h²] / h = 2x + h, and as h → 0, the result is 2x.

例如，若 f(x) = x²，展开 f(x + h) = x² + 2xh + h²，得到 [2xh + h²] / h = 2x + h，当 h → 0 时结果就是 2x。

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3. Power Rule and Basic Rules | 幂法则与基本法则

The power rule is the most frequently used differentiation rule. For any real constant n,

幂法则是最常用的求导法则。对于任意实常数 n，有

d/dx (xⁿ) = n xⁿ⁻¹

This means you multiply by the original power and then reduce the power by 1. The same rule applies to negative and fractional powers after rewriting roots and fractions.

这意味着先乘以原来的指数，再把指数减 1。将根式和分式改写为负指数和分数指数后，同样的法则仍然适用。

You must also know the constant multiple rule d/dx [a f(x)] = a f'(x) and the sum rule d/dx [f(x) + g(x)] = f'(x) + g'(x). These allow term-by-term differentiation of polynomials.

你还需要掌握常数倍法则 d/dx [a f(x)] = a f'(x) 以及加法法则 d/dx [f(x) + g(x)] = f'(x) + g'(x)。这些法则允许我们逐项对多项式求导。

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4. Product Rule | 乘积法则

When a function is the product of two functions, y = u(x) v(x), the derivative is given by:

当一个函数是两个函数的乘积 y = u(x) v(x) 时，导数由下式给出：

dy/dx = u dv/dx + v du/dx

In words: differentiate the first function, leave the second, then add the first multiplied by the derivative of the second. It is often helpful to label u and v clearly before differentiating.

用文字表述：先对第一个函数求导并保留第二个，再加上第一个乘以第二个的导数。在求导之前清晰地标出 u 和 v 通常会很有帮助。

For example, if y = x² sin x, let u = x² and v = sin x; then dy/dx = 2x sin x + x² cos x.

例如，若 y = x² sin x，令 u = x²、v = sin x；则 dy/dx = 2x sin x + x² cos x。

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5. Quotient Rule | 商法则

For a function written as a quotient y = u/v, the derivative is:

对于写成商形式的函数 y = u/v，其导数为：

dy/dx = (v du/dx − u dv/dx) / v²

The numerator starts with the denominator multiplied by the derivative of the numerator. A common memory aid is ‘v du minus u dv over v squared’. Be careful with the subtraction sign.

分子以分母乘以分子的导数开始。一个常见记忆口诀是“v du 减 u dv 除以 v 的平方”。注意不要弄错减号。

Example: if y = x / (x + 1), then u = x, v = x + 1, so dy/dx = [(x + 1)(1) − x(1)] / (x + 1)² = 1 / (x + 1)².

示例：若 y = x / (x + 1)，则 u = x，v = x + 1，因此 dy/dx = [(x + 1)(1) − x(1)] / (x + 1)² = 1 / (x + 1)²。

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6. Chain Rule | 链式法则

The chain rule is used for composite functions, where one function is applied inside another. If y = f(u) and u = g(x), then:

链式

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