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

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

Differentiation is one of the central pillars of Edexcel A-Level Pure Mathematics. It appears in almost every paper, both as a standalone skill and inside applied problems such as optimisation, rates of change, tangents, normals and curve sketching. This revision guide works through the exact techniques you need to master, with the notation and structure expected in Edexcel mark schemes.

微分是 Edexcel A-Level 纯数学的核心支柱之一。它几乎出现在每一份试卷中，既作为独立技能考查，也融入优化、变化率、切线、法线和曲线草图等应用问题。本复习指南将按 Edexcel 评分标准所要求的符号和结构，逐项讲解你必须掌握的技术。

1. First Principles and the Derivative | 第一性原理与导数

The derivative f'(x) gives the instantaneous rate of change of f(x) with respect to x. In Edexcel A-Level, the first-principles definition is most often tested on simple polynomials such as x² or x³.

导数 f'(x) 表示 f(x) 关于 x 的瞬时变化率。在 Edexcel A-Level 考试中，第一性原理定义最常用于考查 x² 或 x³ 等简单多项式。

f'(x) = limₕ→0 [f(x+h) – f(x)] / h

This limit calculates the gradient of the chord between two nearby points and then shrinks the gap to zero. You should be able to expand brackets, cancel terms and take the limit confidently.

该极限计算两个相邻点之间弦的斜率，然后将间隔缩小到零。你应当能够熟练展开括号、约去项并求极限。

For f(x) = x², expanding (x+h)² gives x² + 2xh + h², so the derivative is 2x. | 对于 f(x) = x²，展开 (x+h)² 得到 x² + 2xh + h²，因此导数为 2x。
For f(x) = x³, expanding (x+h)³ gives x³ + 3x²h + 3xh² + h³, so the derivative is 3x². | 对于 f(x) = x³，展开 (x+h)³ 得到 x³ + 3x²h + 3xh² + h³，因此导数为 3x²。

2. Basic Rules and Standard Derivatives | 基本法则与标准导数

Edexcel expects you to use the power rule, constant multiple rule and sum rule fluently before attempting harder questions. These rules allow you to differentiate any polynomial and most simple algebraic functions.

Edexcel 要求你在处理更难的题目之前，能够流利使用幂法则、常数倍法则和和差法则。这些法则使你能够对任意多项式以及大多数简单的代数函数求导。

Power rule: d/dx (xⁿ) = n xⁿ⁻¹ | 幂法则：d/dx (xⁿ) = n xⁿ⁻¹
Constant multiple rule: d/dx [cf(x)] = c f'(x) | 常数倍法则：d/dx [cf(x)] = c f'(x)
Sum and difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x) | 和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)

A standard derivative table is useful for quick recall in the exam. The following table shows the most common results required by Edexcel Pure Mathematics.

标准导数表有助于在考试中快速回忆。下表列出了 Edexcel 纯数学要求的最常见结果。

f(x)	f'(x)
xⁿ	n xⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	-sin x
tan x	sec² x

3. The Chain Rule | 链式法则

The chain rule is used to differentiate composite functions, where one function is applied inside another. Edexcel often writes this as dy/dx = dy/du × du/dx, so it is worth practising that exact notation.

链式法则用于对复合函数求导，即一个函数嵌套在另一个函数内部。Edexcel 通常将其写作 dy/dx = dy/du × du/dx，因此值得练习这种确切的写法。

dy/dx = dy/du × du/dx

For example, if y = (3x² + 1)⁵, let u = 3x² + 1. Then y = u⁵, du/dx = 6x and dy/du = 5u⁴. Multiplying gives dy/dx = 30x(3x² + 1)⁴.

例如，如果 y = (3x² + 1)⁵，设 u = 3x² + 1。则 y = u⁵，du/dx = 6x，dy/du = 5u⁴。相乘得到 dy/dx = 30x(3x² + 1)⁴。

You should recognise chain-rule forms such as [f(x)]ⁿ, eᶠ⁽ˣ⁾, ln(f(x)), sin(f(x)) and cos(f(x)). The derivative is always the derivative of the outer function, evaluated at the inner function, multiplied by the derivative of the inner function.

你应当识别链式法则的常见形式，如 [f(x)]ⁿ、eᶠ⁽ˣ⁾、ln(f(x))、sin(f(x)) 和 cos(f(x))。其结果总是外函数在内部函数处求导，再乘以内部函数的导数。

d/dx [f(x)]ⁿ = n [f(x)]ⁿ⁻¹ f'(x) | d/dx [f(x)]ⁿ = n [f(x)]ⁿ⁻¹ f'(x)
d/dx eᶠ⁽ˣ⁾ = f'(x) eᶠ⁽ˣ⁾ | d/dx eᶠ⁽ˣ⁾ = f'(x) eᶠ⁽ˣ⁾
d/dx ln(f(x)) = f'(x) / f(x) | d/dx ln(f(x)) = f'(x) / f(x)

4. The Product Rule | 乘积法则

When differentiating a product of two functions, you cannot simply multiply the individual derivatives. Edexcel requires the product rule in the exact form u’v + uv’.

当对两个函数的乘积求导时，不能简单地将各自的导数相乘。Edexcel 要求使用乘积法则，其精确形式为 u’v + uv’。

d/dx (uv) = u’v + uv’

A typical exam question is to differentiate y = x² eˣ. Let u = x² and v = eˣ, so u’ = 2x and v’ = eˣ. Therefore dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x).

一个典型的考题是对 y = x² eˣ 求导。设 u = x²，v = eˣ，则 u’ = 2x，v’ = eˣ。因此 dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x)。

It is good practice to factorise the final answer when possible, as this often helps with finding stationary points or interpreting the gradient function.

尽可能对最终答案进行因式分解是一个好习惯，因为这会帮助后续求驻点或解释导函数。

5. The Quotient Rule | 商法则

The quotient rule is needed when one function is divided by another. Edexcel will accept the formula as long as it is applied correctly, but the order of terms in the numerator is essential.

当一个函数除以另一个函数时，需要使用商法则。只要正确应用，Edexcel 会接受公式，但分子中各项的顺序至关重要。

d/dx (u/v) = (v u’ – u v’) / v²

For example, if y = sin x / x, let u = sin x and v = x. Then u’ = cos x, v’ = 1, so dy/dx = [x cos x – sin x] / x².

例如，如果 y = sin x / x，设 u = sin x，v = x。则 u’ = cos x，v’ = 1，所以 dy/dx = [x cos x – sin x] / x²。

Many students lose marks by writing u’v – uv’ instead of v u’ – u v’. A simple memory aid is to keep the denominator function first in the numerator and then square the denominator.

许多学生因为把分子写成 u’v – uv’ 而不是 v u’ – u v’ 而丢分。一个简单的记忆方法是把分母函数放在分子的第一项，然后对分母平方。

6. Differentiating Exponentials and Logarithms | 指数函数与对数函数求导

Exponential and logarithmic functions appear frequently in Edexcel modelling and calculus questions. You must know their derivatives exactly, including the chain-rule versions.

指数函数和对数函数经常出现在 Edexcel 的建模和微积分问题中。你必须准确掌握它们的导数，包括链式法则版本。

d/dx (eˣ) = eˣ, d/dx (ln x) = 1/x for x > 0

For a general exponential base a, the derivative is d/dx (aˣ) = aˣ ln a. Similarly, d/dx (ln kx) = 1/x because the k cancels after applying the chain rule.

对于一般底数 a 的指数函数，导数为 d/dx (aˣ) = aˣ ln a。类似地，d/dx (ln kx) = 1/x，因为在应用链式法则后 k 会约去。

d/dx (eᵏˣ) = k eᵏˣ | d/dx (eᵏˣ) = k eᵏˣ
d/dx (e⁻ˣ) = -e⁻ˣ | d/dx (e⁻ˣ) = -e⁻ˣ
d/dx (ln(ax + b)) = a / (ax + b) | d/dx (ln(ax + b)) = a / (ax + b)

These results are particularly useful when solving differential equations or finding gradients of logarithmic curves.

这些结果在求解微分方程或求对数曲线的斜率时特别有用。

7. Differentiating Trigonometric Functions | 三角函数求导

Trigonometric differentiation in Edexcel Pure Mathematics focuses mainly on sin, cos and tan, but you also need to handle multiples of x and phase shifts using the chain rule.

Edexcel 纯数学中的三角函数求导主要集中在 sin、cos 和 tan，但你还需要使用链式法则处理 x 的倍数和相位平移。

d/dx (sin x) = cos x, d/dx (cos x) = -sin x, d/dx (tan x) = sec² x

For example, d/dx (sin 3x) = 3 cos 3x, because the chain rule multiplies by the coefficient of x. Likewise, d/dx (cos(2x + π/2)) = -2 sin(2x + π/2).

例如，d/dx (sin 3x) = 3 cos 3x，因为链式法则要乘以 x 的系数。同样，d/dx (cos(2x + π/2)) = -2 sin(2x + π/2)。

You should also recognise that sec² x is one of the standard derivatives, and that it can be written as 1/cos² x if required by the mark scheme.

你还应当认识到 sec² x 是一个标准导数，如果评分标准需要，它可以写作 1/cos² x。

8. Parametric Differentiation | 参数微分

When a curve is given in parametric form, x = f(t) and y = g(t), the gradient dy/dx is found by dividing dy/dt by dx/dt. This is a favourite Edexcel topic because it tests chain-rule understanding and algebraic simplification.

当曲线以参数形式给出时，x = f(t)，y = g(t)，斜率 dy/dx 通过 dy/dt 除以 dx/dt 得到。这是 Edexcel 偏爱的考点，因为它考查链式法则的理解和代数化简。

dy/dx = (dy/dt) ÷ (dx/dt)

For instance, if x = t² + 1 and y = t³ – 2t, then dx/dt = 2t and dy/dt = 3t² – 2. The gradient is therefore dy/dx = (3t² – 2) / (2t).

例如，若 x = t² + 1 且 y = t³ – 2t，则 dx/dt = 2t，dy/dt = 3t² – 2。因此斜率为 dy/dx = (3t² – 2) / (2t)。

For the second derivative in parametric form, use d²y/dx² = d/dt [dy/dx] ÷ dx/dt. You must differentiate the expression for dy/dx with respect to t before dividing by dx/dt.

对于参数形式的二阶导数，使用 d²y/dx² = d/dt [dy/dx] ÷ dx/dt。你必须先对 dy/dx 关于 t 求导，然后再除以 dx/dt。

9. Implicit Differentiation | 隐函数微分

Implicit differentiation is used when y is not expressed explicitly in terms of x. Edexcel requires you to differentiate both sides of an equation with respect to x, treating y as a function of x and multiplying by dy/dx whenever differentiating y.

当 y 没有显式表示为 x 的函数时，需要使用隐函数微分。Edexcel 要求你对等式两边关于 x 求导，将 y 视为 x 的函数，并且每当对 y 求导时乘以 dy/dx。

d/dx (yⁿ) = n yⁿ⁻¹ dy/dx

For the equation x² + xy + y² = 7, differentiate term by term: 2x + (y + x dy/dx) + 2y dy/dx = 0. Then collect dy/dx terms to get dy/dx = -(2x + y) / (x + 2y).

对于方程 x² + xy + y² = 7，逐项求导：2x + (y + x dy/dx) + 2y dy/dx = 0。然后合并含 dy/dx 的项，得到 dy/dx = -(2x + y) / (x + 2y)。

Always check that you have applied the product rule to mixed terms such as xy. A common error is to forget the y term when differentiating xy with respect to x.

务必检查你是否对 xy 之类的混合项应用了乘积法则。一个常见错误是在对 xy 关于 x 求导时遗漏 y 项。

10. Second Derivatives and Stationary Points | 二阶导数与驻点

The second derivative d²y/dx² is the derivative of dy/dx. Edexcel uses it to classify stationary points and to investigate concavity.

二阶导数 d²y/dx² 是 dy/dx 的导数。Edexcel 用它来对驻点分类并研究曲线的凹凸性。

d²y/dx² > 0 ⇒ local minimum, d²y/dx² < 0 ⇒ local maximum

For y = x³ – 3x, the first derivative is dy/dx = 3x² – 3. Setting this to zero gives x = ±1. The second derivative d²y/dx² = 6x is positive at x =

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