📚 A-Level Edexcel Pure Maths: Core Revision on Differentiation and Integration | Edexcel A-Level 数学核心复习：微分与积分
In Edexcel A-Level Pure Mathematics, differentiation and integration form the backbone of calculus. This revision guide focuses on the key techniques and exam-style applications that frequently appear in Papers 1 and 2. Mastering these core ideas will help you tackle gradient problems, optimisation, area calculations and kinematics with confidence.
在 Edexcel A-Level 纯数学中，微分与积分构成微积分的核心。本复习指南聚焦试卷一和试卷二中经常出现的关键技巧与考试型应用。掌握这些核心概念将帮助你自信地应对梯度问题、最优化、面积计算和运动学问题。
1. Understanding the Gradient Function | 理解梯度函数
The derivative of a function measures how quickly the function changes at any given point. Geometrically, it gives the gradient of the tangent to the curve y = f(x). For a curve, the gradient is not constant, so we use the derivative f'(x) to describe the instantaneous rate of change.
函数的导数衡量函数在任意给定点的变化快慢。从几何上看，它给出了曲线 y = f(x) 的切线斜率。对于曲线而言，斜率不是常数，因此我们用导数 f'(x) 来描述瞬时变化率。
Gradient of tangent = f'(x) = dy/dx
You should be comfortable using all three notations: f'(x), dy/dx and d/dx[f(x)]. In Edexcel exams, questions may switch between these forms, especially in contextual problems such as motion or rates of change.
你应该熟练使用三种记号：f'(x)、dy/dx 和 d/dx[f(x)]。在 Edexcel 考试中，题目可能会在这些形式之间切换，尤其是在运动或变化率等情境题中。
2. Differentiation from First Principles | 从第一性原理求导
Differentiation from first principles uses the limit definition of the derivative. It is a standard proof that appears regularly in Edexcel papers, often asking you to differentiate a simple quadratic or linear function from first principles.
从第一性原理求导使用导数的极限定义。这是 Edexcel 试卷中经常出现的标准证明，通常要求你对简单的二次函数或线性函数从第一性原理求导。
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
To answer such a question, substitute f(x + h) and f(x) into the formula, expand and simplify, then let h tend to 0. Make sure you write the limit correctly and show each step clearly to earn full marks.
解答此类问题时，将 f(x + h) 和 f(x) 代入公式，展开并化简，然后令 h 趋于 0。确保正确写出极限符号，并清晰展示每一步以获得满分。
3. Rules of Differentiation | 求导法则
For most functions, we use standard differentiation rules rather than first principles. For a power function, bring the power down and subtract one from the exponent. This works for positive, negative and fractional powers before differentiating.
对于大多数函数，我们使用标准求导法则而不是第一性原理。对于幂函数，将指数下移并将指数减一。这适用于正指数、负指数和分数指数函数。
If y = xⁿ then dy/dx = n xⁿ⁻¹
The sum rule allows you to differentiate term by term. For example, if y = 4x³ − 2x + 5, then dy/dx = 12x² − 2. Constants differentiate to zero, which is a common error to avoid.
和法则允许你逐项求导。例如，如果 y = 4x³ − 2x + 5，则 dy/dx = 12x² − 2。常数求导为零，这是需要避免的常见错误。
| y = xⁿ → dy/dx = n xⁿ⁻¹ | 幂函数求导法则 Power rule |
| y = k f(x) → dy/dx = k f'(x) | 常数倍法则 Constant multiple rule |
| y = f(x) + g(x) → dy/dx = f'(x) + g'(x) | 和法则 Sum rule |
4. Tangents and Normals | 切线与法线
Once you have the derivative, you can find the equation of a tangent or normal at a point. The tangent has gradient m = f'(a) at x = a. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal.
一旦求出导数，就可以求出某点处切线和法线的方程。在 x = a 处，切线的斜率 m = f'(a)。法线垂直于切线，因此其斜率是切线斜率的负倒数。
Tangent: y − y₁ = m(x − x₁) and Normal: mₙ = −1 / m
Always find the y-coordinate by substituting into the original function before using the line equation. Edexcel mark schemes expect the final answer in a simplified form, usually ax + by + c = 0 or y = mx + c.
在使用直线方程之前，一定要将 x 代入原函数求出 y 坐标。Edexcel 评分标准要求最终答案化为简化形式，通常是 ax + by + c = 0 或 y = mx + c。
5. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where dy/dx = 0. They can be classified as local maxima, local minima or points of inflection. To classify them, use the second derivative or test the sign of dy/dx on either side of the point.
驻点出现在 dy/dx = 0 处。它们可以分为局部最大值、局部最小值或拐点。要判断其类型，可以使用二阶导数，或检验驻点两侧 dy/dx 的符号。
d²y/dx² > 0 ⇒ minimum; d²y/dx² < 0 ⇒ maximum
When sketching a curve, include the intercepts, stationary points and behaviour as x tends to positive or negative infinity. Students often lose marks by omitting the coordinates of key points or mislabelling axes.
在画曲线草图时，要标出截距、驻点以及 x 趋于正无穷或负无穷时的变化趋势。学生常因遗漏关键点坐标或坐标轴标注错误而失分。
6. Introduction to Integration | 积分入门
Integration is the reverse process of differentiation. Given a derivative dy/dx, integration finds the original function y, up to an unknown constant. This is why an indefinite integral must always include + c.
积分是微分的逆过程。给定导数 dy/dx，积分可求出原函数 y，但会相差一个未知常数。这就是为什么不定积分总是要加上 + c。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + c, n ≠ −1
The rule for integrating powers is the inverse of the differentiation rule: add one to the exponent and divide by the new exponent. Remember that n cannot be −1 in this formula because division by zero is undefined.
幂函数积分法则是求导法则的逆过程：指数加一，再除以新指数。请记住在此公式中 n 不能为 −1，因为除以零无定义。
7. Finding Areas Under Curves | 计算曲线下方面积
A definite integral gives the signed area between a curve and the x-axis. If the curve lies above the x-axis over an interval, the integral is positive. If the curve lies below the axis, the integral is negative.
定积分给出曲线与 x 轴之间的有符号面积。如果在某区间内曲线位于 x 轴上方，则积分为正。如果曲线位于 x 轴下方，则积分为负。
Area = ∫ₐᵇ f(x) dx = F(b) − F(a)
To find the total area when the curve crosses the x-axis, split the interval into sections where the function is positive and negative, then take the absolute value of each area before adding.
当曲线穿过 x 轴时，要求总面积，需要将区间分成函数为正和为负的部分，然后对每部分面积取绝对值后再相加。
8. Definite and Indefinite Integrals | 定积分与不定积分
An indefinite integral produces a family of functions and includes the constant c. A definite integral has upper and lower limits and produces a numerical value. In Edexcel exams, you must show substitution of limits clearly.
不定积分产生一族函数，并包含常数 c。定积分有上限和下限，并产生一个数值。在 Edexcel 考试中，你必须清晰展示代入上下限的过程。
∫ 2x dx = x² + c and ∫₀² 2x dx = [x²]₀² = 4 − 0 = 4
Be careful with functions that require simplification before integration, such as expanding brackets or writing square roots as fractional powers. Always write expressions in index form before applying the power rule.
对于积分前需要化简的函数要小心，例如展开括号或将平方根写成分数指数。在应用幂法则之前，一定要将表达式写成指数形式。
9. Reverse Differentiation and the Constant of Integration | 反向微分与积分常数
Because integration is reverse differentiation, you can check your answer by differentiating it. If you differentiate your result and get back the original integrand, your integration is correct. This is a useful exam technique.
由于积分是微分的逆运算，你可以通过求导来检查答案。如果对结果求导后得到原来的被积函数，则积分正确。这是一个非常有用的考试技巧。
The constant c represents the unknown vertical shift of the original function. In context questions, a boundary condition or initial value is often given to determine c. For example, if y = 2 when x = 0, substitute these values into your integrated expression.
常数 c 代表原函数未知的垂直平移量。在情境题中，通常会给出边界条件或初值来确定 c。例如，若 x = 0 时 y = 2，则将这些值代入积分后的表达式。
dy/dx = 3x² ⇒ y = x³ + c; given y(0) = 5 ⇒ c = 5
10. Applications to Kinematics | 运动学应用
In kinematics, displacement s, velocity v and acceleration a are connected by differentiation and integration. Velocity is the derivative of displacement with respect to time, and acceleration is the derivative of velocity.
在运动学中，位移 s、速度 v 和加速度 a 通过微分和积分相互联系。速度是位移对时间的导数，加速度是速度对时间的导数。
v = ds/dt, a = dv/dt, s = ∫ v dt
Exam questions often give a velocity function and ask for displacement over a time interval, or give acceleration and ask for velocity. Remember that displacement is a vector quantity, so direction matters.
考试题目常给出速度函数并要求求某时间段内的位移，或给出加速度并要求求速度。请记住位移是矢量，因此方向很重要。
11. Common Exam Mistakes | 常见考试错误
One common mistake is forgetting to add the constant c in indefinite integration. Another is misapplying the power rule when n = −1, because the integral of 1/x is the natural logarithm function, not a power.
一个常见错误是在不定积分中忘记加上常数 c。另一个错误是在 n = −1 时误用幂法则，因为 1/x 的积分是自然对数函数，而不是幂函数。
∫ (1/x) dx = ln |x| + cPublished by TutorHao | A-Level Mathematics Revision Series | aleveler.com
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