📚 Mastering Differentiation for Edexcel A-Level Pure Maths | 精通微分：Edexcel A-Level 纯数学
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It underpins rates of change, gradients of curves, optimisation, kinematics and mathematical modelling, and it appears in almost every paper you will sit. A confident grasp of the rules is not optional - it is the foundation on which sequences, series, integration and mechanics questions are built.
微分是 Edexcel A-Level 纯数学中考查频率最高的主题之一。它是变化率、曲线斜率、最优化、运动学以及数学建模等概念的基础，几乎出现在你参加的每一份试卷中。熟练掌握求导法则不是可选项，而是数列、级数、积分和力学题目的共同地基。
This revision guide covers the essential definitions, notation, rules, applications and common pitfalls you must master for Papers 1 and 2. Each section pairs the formal statement with worked reasoning, so you can move from understanding to exam performance.
本复习指南涵盖你必须掌握的核心定义、符号、法则、应用以及常见失分点，帮助你应对 Paper 1 和 Paper 2。每一节都把形式化表述与解题推理配对呈现，让你从"理解"顺利过渡到"得分"。
1. Definition from First Principles | 第一原理定义
In Edexcel Pure Maths, the derivative of a function f(x) is defined as the limit of the average gradient between two points on the curve as the gap h tends to zero. This formal limit is the reason differentiation produces an instantaneous rate of change rather than an average one over an interval.
在 Edexcel 纯数学中，函数 f(x) 的导数定义为曲线上两点之间的平均斜率在间距 h 趋于 0 时的极限。正是这个形式化极限，使得微分给出的是瞬时变化率，而不是某个区间上的平均变化率。
f′(x) = lim (h → 0) [ f(x + h) − f(x) ] / h
ng point x approaching it. Examiners sometimes ask for this version when the question says "differentiate from first principles at the point where x = a".
等价形式是固定一点 a，让动点 x 趋近于它。当题干说"在 x = a 处用第一原理求导"时，考官往往想要的就是这个版本。
f′(a) = lim (x → a) [ f(x) − f(a) ] / (x − a)
− x² = 2xh + h². Dividing by h gives 2x + h, and as h → 0 the limit is 2x, so f′(x) = 2x.
例如，若 f(x) = x²，则 f(x + h) − f(x) = (x + h)² − x² = 2xh + h²。除以 h 得到 2x + h，当 h → 0 时极限为 2x，因此 f′(x) = 2x。
In the exam, first-principles questions are almost always on polynomials, and you are usually told which function to use. The marks are for the algebra of expanding, cancelling h and then taking the limit - so write every step clearly.
考试中的第一原理题几乎总是多项式，而且通常会指定函数。分数给的是展开代数的过程、约去 h 的步骤以及最后取极限的动作，因此每一步都要写清楚。
2. Notation and Key Terminology | 符号与关键术语
Edexcel exam papers frequently switch between Lagrange notation f′(x) and Leibniz notation dy/dx. You must be able to read both, and you should know that d/dx is an operator that means "differentiate with respect to x".
Edexcel 试卷中常交替使用拉格朗日记号 f′(x) 和莱布尼茨记号 dy/dx。你必须能读懂两者，并知道 d/dx 是一个算子，表示"对 x 求导"。
written as y′, f′(x), dy/dx, or d/dx [f(x)]. The second derivative is written f″(x) or d²y/dx², and it tells you how the gradient itself is changing.
若 y = f(x)，则导数可以写成 y′、f′(x)、dy/dx 或 d/dx [f(x)]。二阶导数写作 f″(x) 或 d²y/dx²，它描述的是斜率本身如何变化。
= 3, while dy/dx evaluated at x = 3 is often written with a vertical bar and a subscript.
在特定点求导数值时，要使用正确的记号。f′(3) 表示 x = 3 时的斜率值，而 dy/dx 在 x = 3 处的值常用竖线加下标的方式表示。
- f′(x), y′, dy/dx - first derivative, gradient function.
- f″(x), d²y/dx² - second derivative, rate of change of gradient.
- d/dx - the differentiation operator applied to whatever follows it.
- "Differentiate with respect to x" - the variable in the denominator is x.
- f′(x)、y′、dy/dx - 一阶导数，即斜率函数。
- f″(x)、d²y/dx² - 二阶导数，即斜率的变化率。
- d/dx - 微分算子，作用于紧随其后的表达式。
- "对 x 求导" - 分母中的变量就是 x。
3. Differentiating Powers of x | x 的幂的微分
ive powers, negative powers and fractional powers alike, which means it covers roots and reciprocals too.
A-Level 微分中最常用的法则是幂法则。对于任何实数常量 n，若 f(x) = xⁿ，则 f′(x) = n xⁿ⁻¹。它同时适用于正幂、负幂和分数幂，因此根式和倒数也都能处理。
d/dx (xⁿ) = n xⁿ⁻¹
The rule works with all real powers. For example, x⁵ differentiates to 5x⁴, x⁻³ differentiates to −3x⁻⁴, and x1/2 differentiates to ½ x−1/2.
该法则对所有实数幂都成立。例如 x⁵ 求导得 5x⁴，x⁻³ 求导得 −3x⁻⁴，x1/2 求导得 ½ x−1/2。
Before differentiating anything containing a root or a fraction, rewrite it as a power of x. This single habit prevents most lost marks in Paper 1.
在对含根式或分式的表达式求导前，先把它改写为 x 的幂。这个习惯本身就能避免 Paper 1 中大部分的失分。
- √x = x1/2, so d/dx (√x) = ½ x−1/2 = 1 / (2√x).
- 1/x = x⁻¹, so d/dx (1/x) = −x⁻² = −1/x².
- 1/x² = x⁻², so d/dx (1/x²) = −2x⁻³ = −2/x³.
- ∛x = x1/3, so d/dx (∛x) = (1/3) x−2/3.
- √x = x1/2，所以 d/dx (√x) = ½ x−1/2 = 1 / (2√x)。
- 1/x = x⁻¹，所以 d/dx (1/x) = −x⁻² = −1/x²。
- 1/x² = x⁻²，所以 d/dx (1/x²) = −2x⁻³ = −2/x³。
- ∛x = x1/3，所以 d/dx (∛x) = (1/3) x−2/3。
When you finish, it is good practice to convert negative and fractional powers back into root or fraction form if the question uses that style. Edexcel mark schemes usually accept either form, but a tidy final answer reduces the risk of a misread.
做完之后，如果原题使用的是根式或分式写法，最好把负幂和分数幂换回去。Edexcel 评分标准通常两种形式都接受，但整洁的最终答案能降低被判读错误的风险。
4. Constants, Sums and Polynomials | 常数、和与多项式
Differentiation is linear, which means you can differentiate a sum term by term and pull constant factors out. The derivative of any constant is zero, because a horizontal line has gradient zero everywhere.
微分是线性的，这意味着你可以逐项求导，并把常数因子提出来。任何常数的导数都是零，因为水平线处处斜率为零。
d/dx (c) = 0 d/dx [ a·f(x) ± b·g(x) ] = a·f′(x) ± b·g′(x)
+ 2x − 9. Differentiate each term in turn: 12x² − 14x + 2, and the constant −9 disappears.
这两条性质让多项式求导变成机械流程。以 y = 4x³ − 7x² + 2x − 9 为例，逐项求导得 12x² − 14x + 2，常数项 −9 消失。
- Expand brackets first if it is quicker than using the product rule.
- Collect like terms before differentiating when possible.
- Watch the sign in front of every term - most errors here are sign errors.
- 如果展开括号比用乘积法则更快，就先展开。
- 能合并同类项时先合并，再求导。
- 注意每一项前面的符号 - 这里绝大多数错误都是符号错误。
A typical exam instruction is "find the gradient of the curve at the point where x = 2" ng into the wrong function is one of the most common careless mistakes in the whole paper.
常见的考题指令是"求曲线在 x = 2 处的斜率"。求导之后，要把 x = 2 代入 f′(x)，而不是代入 f(x)。代错函数是整份试卷中最常见的粗心错误之一。
5. Tangents, Normals and Gradients | 切线、法线与斜率
the point of contact, you can write the equation of the tangent using y − y₁ = m(x − x₁).
导数给出的是曲线在某点处切线的斜率。一旦得到斜率 m 和切点坐标，就可以用 y − y₁ = m(x − x₁) 写出切线方程。
The normal is the line perpendicular to the tangent at the point of contact. Because perpendicular gradients multiply to −1, the gradient of the normal is −1/m, provided m ≠ 0.
法线是在切点处与切线垂直的直线。由于垂直直线的斜率乘积为 −1，法线的斜率为 −1/m，前提是 m ≠ 0。
mtangent × mnormal = −1 so mnormal = −1 / mtangent
− 2 = 2(x − 3), that is y = 2x − 4. The normal has gradient −½, giving y − 2 = −½(x − 3).
例如，求 y = x² − 4x + 5 在 x = 3 处的切线与法线。这里 dy/dx = 2x − 4，故 x = 3 时斜率为 2。点为 (3, 2)，因此切线为 y − 2 = 2(x − 3)，即 y = 2x − 4。法线斜率为 −½，得到 y − 2 = −½(x − 3)。
- If the question asks where the tangent is parallel to a given line, set f′(x) equal to that line's gradient.
- If it asks where the tangent is parallel to the x-axis, set f′(x) = 0.
- If it asks where the normal is parallel to a given line, set −1/f′(x) equal to that gradient.
- 若题目问切线与某给定直线平行，则令 f′(x) 等于该直线的斜率。
- 若问切线与 x 轴平行，则令 f′(x) = 0。
- 若问法线与某给定直线平行，则令 −1/f′(x) 等于该斜率。
6. Stationary Points and the Second Derivative | 驻点与二阶导数
ordinates, and you must substitute back into f(x) to find the corresponding y-coordinates.
驻点是斜率为零的点，因此要解方程 f′(x) = 0。解出的是横坐标，必须再代回 f(x) 求出对应的纵坐标。
″(x) = 0, the test is inconclusive and you must use a sign table showing the gradient either side.
判断驻点性质可用二阶导数判别法。若该点处 f″(x) > 0，曲线开口向上，该点为极小值点。若 f″(x) < 0，该点为极大值点。若 f″(x) = 0，判别法失效，必须用两侧斜率符号表来判断。
f′(x) = 0 and f″(x) > 0 → local minimum | f′(x) = 0 and f″(x) < 0 → local maximum
vative is 6 > 0, so (1, −2) is a minimum. At x = −1 it is −6 < 0, so (−1, 2) is a maximum.
以 y = x³ − 3x 为例，dy/dx = 3x² − 3 = 0 得 x = ±1，且 d²y/dx² = 6x。x = 1 时二阶导数为 6 > 0，故 (1, −2) 为极小值点。x = −1 时为 −6 < 0，故 (−1, 2) 为极大值点。
- Always give both coordinates of a stationary point, not just x.
- Write a short conclusion: "minimum" or "maximum", with the test shown.
- A change of concavity alone does not guarantee a point of inflection - check the gradient sign change.
- 驻点要写出完整坐标，不能只写 x。
- 写一句简短结论："极小值"或"极大值"，并展示所用判别法。
- 凹凸性改变本身不足以说明是拐点 - 还要检查两侧斜率的符号变化。
7. The Chain Rule | 链式法则
The chain rule handles composite functions - a function inside another function. If y is a function of u and u is a function of x, then the rates multiply.
链式法则用于处理复合函数，即"函数套函数"。若 y 是 u 的函数而 u 是 x 的函数，则变化率相乘。
dy/dx = dy/du × du/dx
In the faster "outside-inside" ). This version is what you should aim to use fluently in the exam, because it saves time.
在更快的"外内形式"中，若 y = [f(x)]ⁿ，则 dy/dx = n [f(x)]ⁿ⁻¹ × f′(x)。考试中应熟练使用这一形式，因为它节省时间。
- y = (3x + 2)⁵ → dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴.
- y = √4x − 1 = (4x − 1)1/2 → dy/dx = ½(4x − 1)−1/2 × 4 = 2 / √4x − 1.
- y = sin(2x) → dy/dx = 2 cos(2x).
- y = e3x² → dy/dx = 6x e3x².
- y = (3x + 2)⁵ → dy/dx = 5(3x + 2)⁴ × 3 = 15(3x + 2)⁴。
- y = √4x − 1 = (4x − 1)1/2 → dy/dx = ½(4x − 1)−1/2 × 4 = 2 / √4x − 1。
- y = sin(2x) → dy/dx = 2 cos(2x)。
- y = e3x² → dy/dx = 6x e3x²。
Forgetting the inner derivative is the most expensive chain rule error. Whenever you differentiate an "inner function" bracket, ask yourself: what is the derivative of what is inside?
忘记乘内层导数是链式法则中最贵重的错误。每次对带括号的"内层函数"求导时，都要自问：括号里面那一项的导数是什么？
8. The Product Rule | 乘积法则
The product rule is used when two functions of x are multiplied together. Write one as u and the other as v, then apply the formula in the exact order shown.
当两个关于 x 的函数相乘时，使用乘积法则。把其中一个设为 u，另一个设为 v，然后严格按公式顺序代入。
y = uv → dy/dx = u (dv/dx) + v (du/dx)
The order does not affect the answer, but a consistent layout reduces errors. A very common exam mistake is writing the rule as u′v′ instead of u′v + uv′.
顺序不影响结果，但保持统一的书写格式能减少错误。考试中极常见的错误是把法则写成 u′v′，而不是 u′v + uv′。
en dy/dx = x² · 3(x + 1)² + (x + 1)³ · 2x = x(x + 1)² [3x + 2(x + 1)] = x(x + 1)²(5x + 2).
例如：求 y = x²(x + 1)³ 的导数。令 u = x²，则 du/dx = 2x；令 v = (x + 1)³，由链式法则得 dv/dx = 3(x + 1)²。于是 dy/dx = x² · 3(x + 1)² + (x + 1)³ · 2x = x(x + 1)² [3x + 2(x + 1)] = x(x + 1)²(5x + 2)。
- Factorise the final answer when the question asks for it, especially for stationary point work.
- Check whether expanding first would be faster - for low powers it often is.
- Keep brackets around the whole of u and v before substituting.
- 若题目要求因式分解，就把最终答案因式分解，尤其在求驻点时会用到。
- 先判断展开是否更快 - 幂次较低时往往确实更快。
- 代入前先用括号把整个 u 和 v 括起来。
9. The Quotient Rule | 商法则
v, then the derivative involves the denominator squared and a subtraction in a fixed order.
当一个函数除以另一个函数时使用商法则。若 y = u / v，则导数中会出现分母的平方，以及一个顺序固定的减法。
y = u / v → dy/dx = [ v (du/dx) − u (dv/dx) ] / v²
x − 3)² = −7 / (x − 3)². The numerator simplifies to a constant, which is a useful check that your algebra is clean.
例如：求 y = (2x + 1) / (x − 3) 的导数。令 u = 2x + 1，则 du/dx = 2；令 v = x − 3，则 dv/dx = 1。于是 x = [ (x − 3)(2) − (2x + 1)(1) ] / (x − 3)² = (2x − 6 − 2x − 1) / (x − 3)² = −7 / (x − 3)²。分子化简为常数，这正好可以用来自查代数是否干净。
- The subtraction order matters: v·u′ − u·v′, never reversed.
- An alternative is to write the quotient as a product with a negative power and use the product rule.
- Never cancel terms across a subtraction sign inside the numerator.
- 减法顺序很重要：v·u′ − u·v′，绝不能颠倒。
- 另一种做法是把商写成含负幂的乘积，再用乘积法则。
- 不要对分子中带减号的各项进行跨项约分。
10. Exponential, Logarithmic and Trigonometric Functions | 指数、对数与三角函数
Beyond powers of x, you must know the standard derivatives of exponential, logarithmic and trigonometric functions, including the versions with a linear inner function.
除了 x 的幂，你还必须掌握指数函数、对数函数和三角函数的标准导数，包括内层为线性函数的版本。
| f(x) | f′(x) |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ekx | k ekx |
| ln x | 1 / x |
| ln(kx + b) | k / (kx + b) |
| sin x | style="border:1
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