📚 Edexcel A-Level Maths: Differentiation and Integration | 爱德思A-Level数学：微分与积分
Differentiation and integration form the backbone of Edexcel A-Level Pure Mathematics. They appear in almost every paper, from tangent equations to area under a curve. This revision guide walks through the core methods, common notation, and exam-style examples you need to master.
微分与积分是爱德思A-Level纯数学的核心内容，几乎出现在每一份试卷中，从切线方程到曲线下方面积。本复习指南将带你系统梳理核心方法、常用符号和考试题型，帮助你稳拿高分。
1. What the Syllabus Expects | 考纲要求
The Edexcel specification tests three main strands: algebraic differentiation, applying derivatives to graphs, and using integration to reverse differentiation or find areas. You must be confident with both dy/dx and ∫ f(x) dx notation.
爱德思考纲主要考查三条主线：代数微分、导数在图像中的应用，以及用积分作为微分的逆运算或求面积。你必须熟练掌握 dy/dx 和 ∫ f(x) dx 两种记号。
You should be able to differentiate polynomials, exponentials, logarithms and trigonometric functions, then use those skills in modelling and optimisation problems.
你应当能够对多项式、指数函数、对数函数和三角函数求导，并能将这些技能用于建模和优化问题。
For integration, the syllabus expects standard results, definite integrals, and area between curves. Both exact values and numerical approximations may be tested.
在积分方面，考纲要求掌握标准结果、定积分以及曲线之间的面积。考题可能要求准确值，也可能要求数值近似。
2. First Principles and the Derivative | 第一性原理与导数
The derivative is defined as the limit of the average rate of change. For y = f(x), the formal definition is:
导数被定义为平均变化率的极限。对于 y = f(x)，正式定义是：
dy/dx = limₕ→₀ [f(x+h) − f(x)] / h
This definition explains why differentiation measures the gradient of a curve at a single point. It links the secant line between two nearby points to the tangent line at one point.
这个定义解释了为什么微分可以度量曲线在某一点的斜率。它将两点之间的割线与某一点的切线联系起来。
In the exam you may be asked to prove the derivative of x² or x³ from first principles, so memorise the expansion steps and the limiting process.
考试中可能要求你用第一性原理证明 x² 或 x³ 的导数，因此要熟记展开步骤和取极限的过程。
3. Differentiation Rules | 微分法则
For polynomial terms, use the power rule: multiply by the power, then reduce the power by one. This is the most frequently used rule in Edexcel Pure Maths.
对于多项式项，使用幂法则：先乘以指数，再将指数减一。这是爱德思纯数学中最常用的法则。
If y = xⁿ, then dy/dx = n xⁿ⁻¹
The chain rule, product rule and quotient rule are essential for composite, product and rational functions. Recognising which rule applies is often half the battle.
链式法则、乘积法则和商法则分别处理复合函数、乘积函数和分式函数。判断该用哪条法则往往已经成功了一半。
Chain rule: dy/dx = dy/du × du/dx
Product rule: d/dx [u v] = u dv/dx + v du/dx
Quotient rule: d/dx [u/v] = (v du/dx − u dv/dx) / v²
Learn to differentiate standard functions quickly: eˣ, ln x, sin x, cos x and tan x all have well-known derivatives that must become automatic.
要熟练掌握标准函数的导数：eˣ、ln x、sin x、cos x 和 tan x 的导数都应成为条件反射。
| Function | Derivative |
| xⁿ | n xⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
4. Tangents, Normals and Rates of Change | 切线、法线与变化率
The gradient of a curve at x = a is given by evaluating dy/dx at that point. The tangent has the same gradient, while the normal has gradient −1 / m.
曲线在 x = a 处的斜率由 dy/dx 在该点的值给出。切线的斜率与之相同，而法线的斜率为 −1 / m。
m_tangent = dy/dx at x = a, m_normal = −1 / m_tangent
Rates of change problems often connect two variables through the chain rule, such as dV/dt = dV/dr × dr/dt. Write down the known rates before substituting numbers.
变化率问题通常通过链式法则联系两个变量，例如 dV/dt = dV/dr × dr/dt。代入数字之前，先列出已知的变化率。
Always write down the point of contact and the gradient before forming y − y₁ = m(x − x₁). This prevents sign errors and shows clear method marks.
在建立 y − y₁ = m(x − x₁) 之前，务必先写出切点坐标和斜率。这样可以避免符号错误，并展示清晰的解题步骤。
5. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where dy/dx = 0. Use the second derivative or a sign test to classify them as maxima, minima or points of inflection.
驻点出现在 dy/dx = 0 处。使用二阶导数或符号检验来判断它们是极大值点、极小值点还是拐点。
d²y/dx² > 0 → minimum; d²y/dx² < 0 → maximum; d²y/dx² = 0 → check sign change
Sketching questions usually ask for intercepts, stationary points and behaviour as x → ±∞. Label every key point clearly on your diagram.
草图题通常要求给出截距、驻点以及 x → ±∞ 时的变化趋势。在图上清楚地标出每一个关键点。
A table of signs for dy/dx is often safer than relying only on the second derivative, especially for repeated roots where d²y/dx² may also be zero.
制作 dy/dx 的符号表通常比只依赖二阶导数更稳妥，尤其是在有重根且 d²y/dx² 也可能为零的情况下。
6. Introduction to Integration | 积分入门
Integration is the reverse process of differentiation. The indefinite integral of xⁿ is xⁿ⁺¹ / (n + 1) + c, provided n ≠ −1.
积分是微分的逆运算。xⁿ 的不定积分为 xⁿ⁺¹ / (n + 1) + c，其中 n ≠ −1。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + c, n ≠ −1
Never forget the constant of integration in an indefinite integral; examiners often deduct a mark for missing ‘+ c’.
不定积分中千万不要漏掉积分常数；漏写 ‘+ c’ 经常会被扣分。
Standard results include ∫ eˣ dx = eˣ + c, ∫ 1/x dx = ln|x| + c, and ∫ cos x dx = sin x + c.
标准结果包括 ∫ eˣ dx = eˣ + c，∫ 1/x dx = ln|x| + c，以及 ∫ cos x dx = sin x + c。
| Function | Integral |
| xⁿ, n ≠ −1 | xⁿ⁺¹ / (n + 1) + c |
| eˣ | eˣ + c |
| 1/x | ln|x| + c |
| sin x | −cos x + c |
| cos x | sin x + c |
| sec² x | tan x + c |
7. Finding Areas under Curves | 曲线下方面积
The area between a curve y = f(x), the x-axis and the lines x = a, x = b is given by the definite integral from a to b of f(x) dx.
曲线 y = f(x)、x 轴以及直线 x = a、x = b 所围成的面积等于 f(x) 在 a 到 b 上的定积分。
Area = ∫ₐᵇ f(x) dx
If the curve lies below the x-axis, the integral is negative, so take the absolute value or split the region into above-axis and below-axis parts.
如果曲线位于 x 轴下方，积分值为负，此时应取绝对值或将该区域分为轴上部分和轴下部分。
For areas between two curves, integrate the difference: Area = ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on [a, b].
求两条曲线之间的面积时，对被积函数之差积分：面积 = ∫ₐᵇ [f(x) − g(x)] dx，其中在 [a, b] 上 f(x) ≥ g(x)。
Always find the points of intersection first, because they determine the limits a and b of the area.
一定要先求出交点，因为交点决定了面积的上下限 a 和 b。
8. Definite vs Indefinite Integrals | 定积分与不定积分
An indefinite integral gives a family of functions and includes ‘+ c’. A definite integral has upper and lower limits and produces a numerical value.
不定积分给出的是一族函数，并带有 ‘+ c’。定积分有上下限，其结果是数值。
∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) − F(a)
Always substitute the
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