Edexcel A-Level Pure Maths: Differentiation Techniques | 爱德思 A-Level 纯数学：微分技巧

📚 Edexcel A-Level Pure Maths: Differentiation Techniques | 爱德思 A-Level 纯数学：微分技巧

Differentiation is a core skill in Edexcel A-Level Pure Mathematics. It allows you to calculate the gradient of a curve, analyse stationary points, construct tangents and normals, and solve real-world modelling problems. This revision guide covers the main differentiation rules and their applications, with paired bilingual explanations to support your study.

微分是爱德思 A-Level 纯数学的核心技能。它帮助计算曲线的斜率、分析驻点、建立切线与法线方程，并解决实际建模问题。本复习指南涵盖主要微分法则及其应用，并提供中英对照讲解以辅助学习。

1. What Differentiation Means | 微分的含义

Differentiation measures how a function changes as its input changes. The derivative f'(x) gives the gradient of the tangent to the curve y = f(x) at any point x.

微分衡量函数随自变量变化的快慢。导数 f'(x) 表示曲线 y = f(x) 在任意点 x 处的切线斜率。

The notation dy/dx is used when y is expressed as a function of x. It represents the instantaneous rate of change of y with respect to x.

当 y 表示为 x 的函数时，使用记号 dy/dx。它表示 y 关于 x 的瞬时变化率。

For example, if y = x², then dy/dx = 2x. This means that at x = 3, the curve has a gradient of 6.

例如，若 y = x²，则 dy/dx = 2x。这表明在 x = 3 处，曲线的斜率为 6。

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2. Differentiation from First Principles | 从第一原理求导

The formal definition of the derivative comes from a limit process. It is often tested at the start of Edexcel Pure Maths courses.

导数的正式定义来自极限过程。爱德思纯数学课程通常在开头考查这一内容。

f'(x) = lim (Δx→0) [f(x+Δx) – f(x)] / Δx

This formula calculates the gradient of a chord as the two points get infinitely close together. The chord becomes the tangent at the point x.

该公式计算两点无限接近时弦的斜率。弦最终变为点 x 处的切线。

To differentiate x² from first principles, expand f(x+Δx) = (x+Δx)² = x² + 2xΔx + (Δx)². Subtract f(x), divide by Δx, and let Δx approach 0 to obtain 2x.

要从第一原理求 x² 的导数，先展开 f(x+Δx) = (x+Δx)² = x² + 2xΔx + (Δx)²。减去 f(x)，除以 Δx，再令 Δx 趋于 0，即得 2x。

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3. Power Rule and Basic Derivatives | 幂函数法则与基本导数

The power rule is the foundation for most differentiation at A-Level. For any real constant n, the derivative of xⁿ is n xⁿ⁻¹.

幂函数法则是 A-Level 中大多数微分运算的基础。对于任意实数常数 n，xⁿ 的导数为 n xⁿ⁻¹。

d/dx (xⁿ) = n xⁿ⁻¹

Constants have zero derivative: d/dx (k) = 0. The derivative of a linear term ax is simply a.

常数的导数为零：d/dx (k) = 0。一次项 ax 的导数就是 a。

Common results include d/dx (sin x) = cos x, d/dx (cos x) = -sin x, and d/dx (eˣ) = eˣ. The natural logarithm gives d/dx (ln x) = 1/x.

常见结果包括 d/dx (sin x) = cos x，d/dx (cos x) = -sin x，以及 d/dx (eˣ) = eˣ。自然对数满足 d/dx (ln x) = 1/x。

• Power rule: multiply by the power and reduce the power by 1 | 幂法则：乘以指数并将指数减 1
• Constant multiple: d/dx [a f(x)] = a f'(x) | 常数倍：d/dx [a f(x)] = a f'(x)
• Sum rule: d/dx [f(x) + g(x)] = f'(x) + g'(x) | 和法则：d/dx [f(x) + g(x)] = f'(x) + g'(x)

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4. Chain Rule | 链式法则

The chain rule is used when one function is inside another. If y = f(g(x)), then multiply the derivative of the outer function by the derivative of the inner function.

链式法则用于一个函数嵌套在另一个函数内部的情形。若 y = f(g(x))，则将外层函数的导数乘以内层函数的导数。

dy/dx = f'(g(x)) g'(x)

For example, if y = (3x² + 5)⁴, let u = 3x² + 5. Then y = u⁴, dy/du = 4u³, and du/dx = 6x. Multiplying gives dy/dx = 24x(3x² + 5)³.

例如，若 y = (3x² + 5)⁴，令 u = 3x² + 5。则 y = u⁴，dy/du = 4u³，du/dx = 6x。相乘得到 dy/dx = 24x(3x² + 5)³。

In Edexcel questions, chain rule appears with trigonometric, exponential, logarithmic and radical functions. Always identify the inner function u before differentiating.

在爱德思试题中，链式法则常与三角函数、指数函数、对数函数和根式函数一起出现。求导前一定要先识别内层函数 u。

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5. Product Rule | 积法则

The product rule is applied when two functions are multiplied together. If y = u v, where u and v are functions of x, then the derivative is u’ v + u v’.

积法则用于两个函数相乘的情形。若 y = u v，其中 u 和 v 都是 x 的函数，则导数为 u’ v + u v’。

d/dx (u v) = u’ v + u v’

A typical example is y = x² sin x. Here u = x², v = sin x, so u’ = 2x and v’ = cos x. Therefore dy/dx = 2x sin x + x² cos x.

典型例子是 y = x² sin x。这里 u = x²，v = sin x，所以 u’ = 2x，v’ = cos x。因此 dy/dx = 2x sin x + x² cos x。

It is often helpful to write u, v, u’ and v’ separately before substituting into the formula. This reduces sign errors and lost terms.

最好先把 u、v、u’ 和 v’ 分别写出，再代入公式。这样可以减少符号错误和漏项。

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6. Quotient Rule | 商法则

The quotient rule is used for division of two functions. If y = u / v, then the derivative is (v u’ – u v’) / v².

商法则用于两个函数相除的情形。若 y = u / v，则导数为 (v u’ – u v’) / v²。

d/dx (u / v) = (v u’ – u v’) / v²

The order in the numerator matters: it is v times u’ minus u times v’, not the reverse. The denominator is the square of the original bottom function.

分子的顺序很重要：是 v 乘以 u’ 减去 u 乘以 v’，而不是反过来。分母是原分母函数的平方。

For y = x / (x² + 1), set u = x and v = x² + 1. Then u’ = 1 and v’ = 2x, giving dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)².

对于 y = x / (x² + 1)，设 u = x，v = x² + 1。则 u’ = 1，v’ = 2x，得到 dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)²。

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7. Tangents and Normals | 切线与法线

After finding dy/dx, you can calculate the gradient of a curve at a given point. The tangent is the straight line that touches the curve at that point with the same gradient.

求出 dy/dx 后，就可以计算曲线在给定点处的斜率。切线是在该点与曲线具有相同斜率的直线。

The normal is perpendicular to the tangent. If the tangent gradient is m, the normal gradient is -1/m, provided m is not zero.

法线垂直于切线。如果切线斜率为 m，则法线斜率为 -1/m，前提是 m 不为零。

To find the equation of a tangent at x = a, first find yₐ = f(a) and m = f'(a). Then use the straight-line form y – yₐ = m(x – a).

要求 x = a 处的切线方程，先求 yₐ = f(a) 和 m = f'(a)。然后代入直线方程 y – yₐ = m(x – a)。

Always check whether the question asks for the equation in the form y = mx + c or in the form ax + by + c = 0. Edexcel often specifies the required form.

一定要注意题目要求方程写成 y = mx + c 还是 ax + by + c = 0 的形式。爱德思考试通常会明确规定所需形式。

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8. Stationary Points and Curve Sketching | 驻点与曲线草图

Stationary points occur where the gradient is zero, so solve dy/dx = 0. These points can be maximum points, minimum points, or points of inflection.

驻点出现在斜率为零的位置，因此要解方程 dy/dx = 0。这些点可能是极大值点、极小值点或拐点。

To classify a stationary point, find the second derivative d²y/dx². If it is positive, the point is a minimum; if negative, it is a maximum. If it is zero, further testing is needed.

要判断驻点类型，可求二阶导数 d²y/dx²。若为正，该点为极小值点；若为负，该点为极大值点；若为零，则需要进一步检验。

You can also use a gradient table, testing the sign of dy/dx just before and just after the stationary point. This is especially useful when the second derivative is zero.

也可以使用斜率表，检验驻点左侧和右侧 dy/dx 的符号。当二阶导数为零时，这种方法尤其有用。

For y = x³ – 3x, dy/dx = 3x² – 3 = 3(x² – 1). Setting this to zero gives x = 1 and x = -1. The second derivative d²y/dx² = 6x shows x = 1 is a minimum and x = -1 is a maximum.

对于 y = x³ – 3x，dy/dx = 3x² – 3 = 3(x² – 1)。令其为零得到 x = 1 和 x = -1。二阶导数 d²y/dx² = 6x 表明 x = 1 为极小值点，x = -1 为极大值点。

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9. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative tells you how the gradient itself is changing. It describes the curvature of a graph.

二阶导数告诉你斜率本身如何变化。它描述了函数图像的弯曲程度。

If d²y/dx² is positive, the curve is concave upward, meaning the gradient is increasing. If it is negative, the curve is concave downward, meaning the gradient is decreasing.

如果 d²y/dx² 为正，曲线向上凹，即斜率在增加。如果为负，曲线向下凹，即斜率在减小。

A point of inflection is where the curve changes concavity. At a stationary point of inflection, both dy/dx = 0 and d²y/dx² = 0, but the sign of the second derivative changes on either side.

拐点是曲线凹凸性改变的位置。在驻点型拐点处，同时有 dy/dx = 0 和 d²y/dx² = 0，但两侧二阶导数的符号发生变化。

Edexcel questions sometimes ask you to prove maximum or minimum values by referring to the sign of d²y/dx². Make sure you state the sign clearly.

爱德思试题有时要求通过说明 d²y/dx² 的符号来证明极大值或极小值。请务必明确写出符号。

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10. Summary Table of Rules | 微分法则总结表

Rule | 法则	Formula | 公式
Power rule | 幂法则	d/dx (xⁿ) = n xⁿ⁻¹
Chain rule | 链式法则	d/dx [f(g(x))] = f'(g(x)) g'(x)
Product rule | 积法则	d/dx (u v) = u’ v + u v’
Quotient rule | 商法则	d/dx (u / v) = (v u’ – u v’) / v²
Exponential | 指数函数	d/dx (eˣ) = eˣ
Natural log | 自然对数	d/dx (ln x) = 1/x
Sine | 正弦函数	d/dx (sin x) = cos x
Cosine | 余弦函数	d/dx (cos x) = -sin x

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11. Common Mistakes | 常见错误

Many students forget to apply the chain rule when differentiating expressions such as sin(3x) or e^(2x). These are not simply sin x or eˣ.

许多学生在求 sin(3x) 或 e^(2x) 这类表达式的导数时，会忘记使用链式法则。它们并不等同于 sin x 或 eˣ 的导数。

Another common error is reversing the product rule or quotient rule order. Always check the question by differentating your answer mentally or substituting a simple value.

另一个常见错误是颠倒积法则或商法则的顺序。务必通过心算求导或代入一个简单值来检查答案。

When finding stationary points, remember to find the y-coordinate as well as the x-coordinate. The question usually asks for the full coordinates.

求驻点时，记得同时求出 y 坐标和 x 坐标。题目通常要求写出完整坐标。

Finally, do not confuse dy/dx with d²y/dx². The first gives gradient; the second gives concavity and helps classify stationary points.

最后，不要混淆 dy/dx 与 d²y/dx²。前者给出斜率，后者给出凹凸性并帮助判断驻点类型。

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12. Exam Tips for Edexcel | 爱德思考试技巧

In Edexcel Pure Mathematics papers, differentiation questions often combine several rules. Start by identifying the overall structure: product, quotient, or composite function.

在爱德思纯数学试卷中，微分题常常综合多个法则。首先要识别整体结构：是积、商还是复合函数。

Show each stage of working clearly. Marks are awarded for the method, not just for the final answer. If you use u or v substitutions, state them explicitly.

每一步解题过程都要清晰展示。分数按方法给，而不只看最终答案。如果使用 u 或 v 代换，请明确写出。

Practise past-paper questions with a timer. Focus on setting up the correct derivative before simplifying, and always return to the question to check the required form.

限时练习历年真题。先正确建立导数表达式再化简，并且一定要回看题目确认答案所需的形式。

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