📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | 精通爱德思 A-Level 纯数学：微分
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It underpins problems on gradients, tangents, normals, optimisation, kinematics and rates of change. This article reviews the core techniques, common pitfalls and exam strategies you need to master.
微分是爱德思 A-Level 纯数学中考查最频繁的主题之一。它是梯度、切线、法线、优化、运动学和变化率等问题的基础。本文梳理核心技巧、常见失分点和考试策略，帮助你彻底掌握。
1. Gradient of a Curve and First Principles | 曲线梯度与第一性原理
The derivative of a function f(x) at a point x = a is defined as the limit of the average rate of change. This definition, known as differentiation from first principles, is a key proof in Edexcel exams.
函数 f(x) 在 x = a 处的导数定义为平均变化率的极限。这个定义称为第一性原理求导，是爱德思考试中的关键证明。
f'(x) = lim Δx→0 [f(x + Δx) – f(x)] / Δx
For example, for f(x) = x², expanding (x + Δx)² gives x² + 2xΔx + (Δx)², so the limit simplifies to 2x. You should be able to reproduce this proof clearly and justify each step.
例如，对于 f(x) = x²，展开 (x + Δx)² 得到 x² + 2xΔx + (Δx)²，因此极限化简为 2x。你应该能够清晰地写出这一证明并说明每一步的理由。
2. Basic Differentiation Rules | 基本微分法则
The power rule is the foundation of most differentiation. For any real constant n, the derivative of xⁿ is n xⁿ⁻¹. You can also differentiate sums, differences and constant multiples term by term.
幂法则是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。你还可以对和、差和常数倍逐项求导。
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| k f(x) | k f'(x) |
| f(x) ± g(x) | f'(x) ± g'(x) |
Always rewrite roots and reciprocals as powers before differentiating. For instance, √x becomes x¹/² and 1/x³ becomes x⁻³.
求导前务必把根式和倒数写成幂的形式。例如，√x 写作 x¹/²，1/x³ 写作 x⁻³。
3. The Chain Rule | 链式法则
The chain rule is used when one function is inside another. If y = f(u) and u = g(x), then the derivative is given by dy/dx = dy/du × du/dx.
链式法则用于一个函数嵌套在另一个函数中的情形。如果 y = f(u) 且 u = g(x)，则导数为 dy/dx = dy/du × du/dx。
dy/dx = dy/du × du/dx
For example, to differentiate y = (3x² + 5)⁴, let u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.
例如，对 y = (3x² + 5)⁴ 求导，设 u = 3x² + 5。则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。
For linear inner functions, a shortcut applies: d/dx [f(ax + b)] = a f'(ax + b). This is very common in Edexcel papers.
对于线性内层函数，有一个常用技巧：d/dx [f(ax + b)] = a f'(ax + b)。这在爱德思试卷中非常常见。
4. The Product and Quotient Rules | 乘法法则与商法则
When two functions are multiplied, use the product rule. If y = u v, where u and v are functions of x, then dy/dx = u dv/dx + v du/dx.
当两个函数相乘时，使用乘法法则。如果 y = u v，其中 u 和 v 都是 x 的函数，则 dy/dx = u dv/dx + v du/dx。
dy/dx = u dv/dx + v du/dx
For example, if y = x² eˣ, then dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2). Write the factorised form when the question asks for a simplified answer.
例如，如果 y = x² eˣ，则 dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2)。当题目要求化简答案时，应写成因式分解形式。
For division, use the quotient rule. If y = u / v, then dy/dx = (v du/dx – u dv/dx) / v².
对于分式函数，使用商法则。如果 y = u / v，则 dy/dx = (v du/dx – u dv/dx) / v²。
dy/dx = (v du/dx – u dv/dx) / v²
For y = (x² + 1) / (3x – 2), let u = x² + 1 and v = 3x – 2. Then dy/dx = [(3x – 2)(2x) – (x² + 1)(3)] / (3x – 2)² = (3x² – 4x – 3) / (3x – 2)².
对于 y = (x² + 1) / (3x – 2)，设 u = x² + 1，v = 3x – 2。则 dy/dx = [(3x – 2)(2x) – (x² + 1)(3)] / (3x – 2)² = (3x² – 4x – 3) / (3x – 2)²。
5. Differentiating Exponentials and Logarithms | 指数与对数函数的微分
The exponential function eˣ is unique because its derivative is itself. More generally, d/dx [e^(kx)] = k e^(kx). This rule appears in growth and decay problems.
指数函数 eˣ 的独特之处在于它的导数等于它本身。更一般地，d/dx [e^(kx)] = k e^(kx)。这一法则出现在增长和衰减问题中。
For natural logarithms, d/dx [ln x] = 1/x for x > 0. Using the chain rule, d/dx [ln(ax + b)] = a/(ax + b), where the expression inside the logarithm must be positive unless absolute values are used.
对于自然对数，当 x > 0 时 d/dx [ln x] = 1/x。使用链式法则，d/dx [ln(ax + b)] = a/(ax + b)，其中对数内部的表达式必须为正，除非使用绝对值。
Example: differentiate y = 5 e^(3x – 1). The derivative is dy/dx = 5 × 3 e^(3x – 1) = 15 e^(3x – 1).
示例：对 y = 5 e^(3x – 1) 求导。导数为 dy/dx = 5 × 3 e^(3x – 1) = 15 e^(3x – 1)。
6. Differentiating Trigonometric Functions | 三角函数的微分
All A-Level candidates must know the standard trigonometric derivatives. They are used directly and also with the chain, product and quotient rules.
所有 A-Level 考生都必须掌握标准三角函数的导数。它们既会直接考查，也会与链式法则、乘法法则和商法则结合考查。
| Function f(x) | Derivative f'(x) |
|---|---|
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
| cosec x | -cosec x cot x |
| sec x | sec x tan x |
| cot x | -cosec² x |
Example: differentiate y = sin(2x + 1). By the chain rule, dy/dx = 2 cos(2x + 1). Example: differentiate y = tan³ x. Let u = tan x, so dy/dx = 3 tan² x × sec² x = 3 tan² x sec² x.
示例：对 y = sin(2x + 1) 求导。由链式法则，dy/dx = 2 cos(2x + 1)。示例：对 y = tan³ x 求导。设 u = tan x，则 dy/dx = 3 tan² x × sec² x = 3 tan² x sec² x。
7. Parametric Differentiation | 参数方程微分
When a curve is given by x = f(t) and y = g(t), the gradient is found by differentiating both with respect to the parameter t. The rule is dy/dx = (dy/dt) / (dx/dt).
当曲线由 x = f(t) 和 y = g(t) 给出时，先对参数 t 分别求导，再使用公式 dy/dx = (dy/dt) / (dx/dt) 求梯度。
dy/dx = (dy/dt) / (dx/dt)
Example: a curve has x = t² – 1 and y = t³ + 2t. Then dx/dt = 2t and dy/dt = 3t² + 2, so dy/dx = (3t² + 2) / (2t). To find the gradient at t = 1, substitute to get dy/dx = (3 + 2) / 2 = 5/2.
示例：曲线参数方程为 x = t² – 1，y = t³ + 2t。则 dx/dt = 2t，dy/dt = 3t² + 2，因此 dy/dx = (3t² + 2) / (2t)。求 t = 1 处的梯度，代入得 dy/dx = (3 + 2) / 2 = 5/2。
8. Implicit Differentiation | 隐函数微分
When a curve is given as an equation involving x and y, you can differentiate both sides with respect to x. Remember that y is a function of x, so d/dx [y] = dy/dx and d/dx [y²] = 2y dy/dx.
当曲线以含 x 和 y 的方程给出时，可以对两边关于 x 求导。记住 y 是 x 的函数，因此 d/dx [y] = dy/dx，d/dx [y²] = 2y dy/dx。
Example: differentiate x² + y² = 25 with respect to x. This gives 2x + 2y dy/dx = 0, so dy/dx = -x/y.
示例：对 x² + y² = 25 两边关于 x 求导。得到 2x + 2y dy/dx = 0，因此 dy/dx = -x/y。
If the equation contains a product such as xy, apply the product rule: d/dx [xy] = x dy/dx + y × 1. This is a frequent source of errors, so check every term carefully.
如果方程中含有 xy 这样的乘积项，要使用乘法法则：d/dx [xy] = x dy/dx + y × 1。这是常见的错误来源，因此要仔细检查每一项。
9. Second Derivatives and Stationary Points | 二阶导数与驻点
The second derivative f”(x) is obtained by differentiating f'(x). It tells you how the gradient is changing and is used to classify stationary points.
二阶导数 f”(x) 是对 f'(x) 再求导得到的。它反映梯度的变化情况，用于判断驻点的性质。
At a stationary point, dy/dx = 0. If f”(a) is positive at x = a, the point is a local minimum. If f”(a) is negative, it is a local maximum. If f”(a) = 0, use the sign of dy/dx on either side of the point to decide its nature.
在驻点处，dy/dx = 0。如果在 x = a 处 f”(a) 为正，则该点为局部极小值；如果 f”(a) 为负，则为局部极大值；如果 f”(a) = 0，则需要考察 dy/dx 在该点两侧的符号来判断性质。
Example: for y = x³ – 3x, dy/dx = 3x² – 3. Setting dy/dx = 0 gives x = ±1. Since f”(x) = 6x, f”(1) = 6, so x = 1 is a minimum; f”(-1) = -6, so x = -1 is a maximum.
示例：对于 y = x³ – 3x，dy/dx = 3x² – 3。令 dy/dx = 0 得 x = ±1。因为 f”(x) = 6x，f”(1) = 6，所以 x = 1 处为极小值；f”(-1) = -6，所以 x = -1 处为极大值。
10. Connected Rates of Change | 相关变化率
In many real-world problems, one quantity changes with respect to time while another quantity changes with respect to the first. You can link these rates using the chain rule.
在许多实际问题中，一个量随时间变化，而另一个量随第一个量变化。你可以用链式法则把这些变化率联系起来。
For example, if the volume V of a sphere increases as its radius r increases, then dV/dt = dV/dr × dr/dt. The formula for the volume of a sphere is V = (4/3)πr³, so dV/dr = 4πr².
例如，如果球体的体积 V 随半径 r 增大而增大，则 dV/dt = dV/dr × dr/dt。球体体积公式为 V = (4/3)πr³，因此 dV/dr = 4πr²。
Always identify the given rate, the required rate and the linking variable before substituting values. Write the chain rule equation first to make your method clear to the examiner.
在代入数值之前，务必先确定已知变化率、待求变化率和连接变量。先写出链式法则方程，让阅卷人清楚地看到你的解题方法。
11. Common Edexcel Exam Pitfalls | 爱德思考试常见失分点
The following mistakes appear every year in Edexcel marking reports. Avoid them by checking your work systematically.
以下错误每年都会出现在爱德思阅卷报告中。通过系统检查来避免这些错误。
- Forgetting the chain rule: d/dx [(2x + 1)⁵] = 10(2x + 1)⁴, not 5(2x + 1)⁴. | 忘记链式法则：d/dx [(2x + 1)⁵] = 10(2x + 1)⁴，而不是 5(2x + 1)⁴。
- Sign errors in the quotient rule: the numerator is v du/dx – u dv/dx, not u dv/dx – v du/dx. | 商法则中的符号错误：分子是 v du/dx – u dv/dx，而不是 u dv/dx – v du/dx。
- Confusing the derivative of sin² x: it is 2 sin x cos x = sin 2x, not cos² x. | 混淆 sin² x 的导数：它是
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