📚 A-Level Edexcel Mathematics: Differentiation and Integration Essentials | A-Level Edexcel 数学：微分与积分核心精讲
Differentiation and integration are two of the most heavily weighted topics in the Edexcel A-Level Mathematics Pure papers. They appear not only as direct skill questions but also inside modelling, optimisation, kinematics and area problems. This guide brings together the key facts, worked-style summaries and exam strategies to help you revise efficiently.
微分与积分是 Edexcel A-Level 数学纯数试卷中权重最高的两大主题。它们不仅以直接计算题出现，还会融入建模、优化、运动学和面积问题。本指南汇集关键知识点、解题思路与考试策略，帮助你高效复习。
1. The Derivative as a Gradient Function | 导数作为斜率函数
The derivative of y = f(x), written f'(x) or dy/dx, gives the gradient of the curve at each point x. It is defined by the limit of the average rate of change as h tends to 0.
函数 y = f(x) 的导数，记作 f'(x) 或 dy/dx，表示曲线在每一点 x 处切线的斜率。它由平均变化率在 h 趋于 0 时的极限定义。
f'(x) = lim(h→0) [f(x + h) − f(x)] / h
Geometrically, the derivative tells you how steep the curve is at a point. If f'(x) > 0 the function is increasing at that point; if f'(x) < 0 the function is decreasing; if f'(x) = 0 the tangent is horizontal and a stationary point may be present.
从几何上看，导数告诉你曲线在某一点的陡峭程度。如果 f'(x) > 0，函数在该点递增；如果 f'(x) < 0，函数在该点递减；如果 f'(x) = 0，切线水平，可能存在驻点。
The second derivative, written f''(x) or d²y/dx², describes how the gradient itself is changing. Edexcel expects you to move fluently between the three notations f'(x), dy/dx and d²y/dx² within the same question.
二阶导数记作 f''(x) 或 d²y/dx²，描述斜率本身的变化方式。Edexcel 要求你在同一道题中能在 f'(x)、dy/dx 与 d²y/dx² 三种记号之间自如切换。
f''(x) = d/dx (dy/dx) = d²y/dx²
2. Differentiation Rules for Polynomials | 多项式的求导法则
For any real power n, the power rule states that the derivative of xⁿ is nxⁿ⁻¹. You can differentiate term by term, and constant multipliers can be carried through the differentiation.
对于任意实数指数 n，幂法则指出 xⁿ 的导数是 nxⁿ⁻¹。你可以逐项求导，常数系数可以保留在求导过程中。
d/dx (xⁿ) = nxⁿ⁻¹
If f(x) = 4x³ − 2x + 7, then f' vative of a constant term is always 0, and the derivative of a linear term ax is simply a.
如果 f(x) = 4x³ − 2x + 7，则 f'(x) = 12x² − 2。常数项的导数永远是 0，一次项 ax 的导数就是 a。
- d/dx [a f(x) ± b g(x)] = a f'(x) ± b g'(x)
- Rewrite roots and reciprocals as powers before differentiating: √x = x1/2, 1/x² = x⁻², 1/√x = x−1/2
- Example: d/dx (3x⁴ − 5/x) = 12x³ + 5x⁻² = 12x³ + 5/x²
Most marks in polynomial differentiation are lost through sign slips and through failing to rewrite a fraction as a negative power before differentiating.
多项式求导丢分最常见的原因是符号错误，以及没有先把分式改写成负指数形式再求导。
3. Tangents, Normals and Rates of Change | 切线、法线与变化率
To find the tangent at the point (x₁, y₁) on y = f(x), evaluate m = f' − x₁). The normal is perpendicular to the tangent, so its gradient is −1/m provided m ≠ 0.
要求 y = f(x) 上点 (x₁, y₁) 处的切线，先求 m = f'(x₁)，再代入 y − y₁ = m(x − x₁)。法线与切线垂直，当 m ≠ 0 时其斜率为 −1/m。
Tangent: y − y₁ = m(x − x₁) Normal: y − y₁ = −(1/m)(x − x₁)
dr/dt). Always write down the chain rule statement explicitly before substituting numbers, because Edexcel awards a method mark for it.
相关变化率问题通过链式法则连接两个量，例如 dV/dt = (dV/dr) × (dr/dt)。代入数值前一定要明确写出链式法则表达式，因为 Edexcel 会为此给出方法分。
- Identify what is given and what is required, then build the chain.
- Substitute the instantaneous values only at the final step.
- Keep units consistent (cm³ s⁻¹, m s⁻¹ and similar).
4. Stationary Points and the Second Derivative | 驻点与二阶导数
A stationary point occurs where f'(x) = 0. To classify it, either examine the sign of f'(x) on both sides, or evaluate the second derivative at the point.
驻点出现在 f'(x) = 0 的地方。判断其性质，可以检查该点两侧 f'(x) 的符号，也可以在该点求二阶导数的值。
| Condition | 条件 | Conclusion | 结论 |
|---|---|
| f'(x) = 0 and f''(x) < 0 | Local maximum | 极大值点 |
| f'(x) = 0 and f''(x) > 0 | Local minimum | 极小值点 |
| f'(x) = 0 and f''(x) = 0 | Test needed (may be a point of inflection) | 需进一步检验（可能是拐点） |
imum may also occur at an endpoint. Global (absolute) extrema questions therefore require you to compare the value of f at every stationary point with the value of f at the endpoints.
在区间 a ≤ x ≤ b 上，最大值或最小值也可能出现在端点。因此求全局极值的问题，需要把每个驻点处的函数值与端点处的函数值进行比较。
- Differentiate, set f'(x) = 0, then solve exactly — leave answers as surds or fractions where sensible.
- Substitute the x-value back into the original equation to get the full coordinate pair.
- Write a conclusion sentence, e.g. "so (2, −5) is a minimum point".
5. Chain, Product and Quotient Rules | 链式、乘积与商法则
When a function is composite, use the chain rule. When it is a product, use the product rule. When it is a quotient, use the quotient rule. Recognising which structure you are facing is the single most important skill in this section.
当函数是复合函数时用链式法则；是乘积时用乘积法则；是商时用商法则。识别题目属于哪种结构，是本部分最重要的能力。
dy/dx = (dy/du) × (du/dx)
d/dx (uv) = u'v + uv'
d/dx (u/v) = (u'v − uv') / v²
t), which is widely used in Edexcel Paper 2. Implicit differentiation extends the same idea: differentiate every term with respect to x, remembering that the derivative of y² is 2y dy/dx.
对于参数方程曲线，链式法则写成 dy/dx = (dy/dt) ÷ (dx/dt)，这在 Edexcel 第二套试卷中很常用。隐函数求导则是同一思想的延伸：对每一项关于 x 求导，注意 y² 的导数是 2y dy/dx。
- x(x + 3)⁵ → product rule with u = x and v = (x + 3)⁵, and v itself needs the chain rule.
- (2x + 1)/(x − 3) → quotient rule, then simplify the numerator fully.
- sin(3x²) → chain rule: 6x cos(3x²).
6. Differentiating Exponential and Trigonometric Functions | 指数函数与三角函数的求导
These standard results must be memorised; they are quoted freely in mark schemes and are rarely given in the formula booklet.
这些标准结果必须背熟；它们常直接出现在评分标准中，而公式册里很少给出。
| Function | 函数 | Derivative | 导数 |
|---|---|
| eˣ | eˣ |
| ekx | k ekx |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec²x |
Trigonometric calculus only works in radians. If a question is set in degrees, the derivative of sin x° is (π/180) cos x°, which is why Edexcel almost always works in radians once calculus begins.
三角函数的微积分只在弧度制下成立。若题目以角度制给出，sin x° 的导数是 (π/180) cos x°，这正是 Edexcel 在进入微积分后几乎总是使用弧度制的原因。
nt rules combine with these results, for example d/dx (x² ln x) = 2x ln x + x, and d/dx (e2x sin x) = e2x(2 sin x + cos x).
链式、乘积与商法则会与这些结果组合使用，例如 d/dx (x² ln x) = 2x ln x + x，以及 d/dx (e2x sin x) = e2x(2 sin x + cos x)。
7. Integration as the Reverse of Differentiation | 积分作为微分的逆运算
Integration reverses differentiation. Because the derivative of any constant is 0, an indefinite integral always carries an arbitrary constant of integration, written + C.
积分是微分的逆运算。由于任何常数的导数都是 0，不定积分总要加上任意积分常数，记作 + C。
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C (n ≠ −1)
∫ (6x² − 4x + 3) dx = 2x³ − 2x² + 3x + C
You can always check an integral by differentiating your answer: if you recover the original integrand, the integration is correct apart from the constant. In an exam this takes seconds and protects you from a whole family of errors.
你随时可以通过对结果求导来检验积分：如果回到原被积函数，说明积分正确（常数除外）。考试中这一步只需几秒，却能避免一整类错误。
- Integrate term by term, exactly as you differentiate term by term.
- Increase the power by 1, then divide by the new power.
- Never forget + C in an indefinite integral — it is a standard accuracy mark.
- Given a point on the curve, substitute it to find the value of C.
8. Standard Integrals and Integration Techniques | 标准积分与积分技巧
Alongside the power rule, you must know the standard integrals of the exponential and trigonometric functions.
除了幂法则，你还必须掌握指数函数与三角函数的标准积分。
∫ eˣ dx = eˣ + C ∫ (1/x) dx = ln|x| + C
∫ sin x dx = −cos x + C ∫ cos x dx = sin x + C ∫ sec²x dx = tan x + C
For linear expressions of the form (ax + b)ⁿ, reverse the chain rule by dividing by the inner derivative a. The same trick handles eax+b and trigonometric functions of ax + b.
对于 (ax + b)ⁿ 形式的线性表达式，可以反向使用链式法则，除以内部导数 a。同样的技巧适用于 eax+b 以及 ax + b 的三角函数。
∫ (ax + b)ⁿ dx = (ax + b)ⁿ⁺¹ / [a(n + 1)] + C
∫ eax+b dx = (1/a) eax+b + C ∫ sin(ax + b) dx = −(1/a) cos(ax + b) + C
Fractional powers, negative powers and surds are all handled by rewriting first: ∫ √x dx becomes ∫ x1/2 dx, giving (2/3)x3/2 + C. The absolute value in ln|x| matters because the integrand 1/x is defined for negative x too.
分数指数、负指数和根式都先改写再积分：∫ √x dx 写成 ∫ x1/2 dx，得到 (2/3)x3/2 + C。ln|x| 中的绝对值符号很重要，因为被积函数 1/x 在 x 为负时同样有定义。
9. Definite Integrals and the Fundamental Theorem | 定积分与微积分基本定理
A definite
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