📚 Edexcel A-Level Maths: Differentiation and Integration Essentials | 爱德思 A-Level 数学：微分与积分核心精讲
Differentiation and integration form the backbone of Edexcel A-Level Pure Mathematics. They appear in almost every paper, from first-principles proofs to optimisation and area problems. This revision guide breaks down the essential methods, common pitfalls, and exam strategies you need to master.
微分与积分是爱德思 A-Level 纯数学的核心内容。从第一原理证明到最优化和面积问题，它们几乎出现在每份试卷中。本复习指南将拆解你必须掌握的核心方法、常见误区和考试策略。
1. Curriculum Map and Command Words | 考纲地图与指令词
Edexcel Pure Mathematics topics tested under the new specification include proof, algebra, functions, coordinate geometry, sequences, trigonometry, exponentials, logarithms, differentiation, integration, and numerical methods. Questions often use command words such as ‘prove’, ‘show that’, ‘find’, ‘hence’, and ‘state’.
新考纲下的爱德思纯数学考点包括证明、代数、函数、坐标几何、数列、三角学、指数与对数、微分、积分和数值方法。题目常使用指令词，如“证明”“求证”“求”“由此”“写出”等。
For differentiation and integration, marks are awarded for method, accuracy, and sometimes clarity of notation. Always show the chain rule steps or the limits of integration clearly.
对于微分和积分，分数通常分配给方法分、答案准确分，有时还包括表述清晰分。务必清晰展示链式法则步骤或积分上下限。
2. Differentiation from First Principles | 第一原理求导
The derivative is defined by the limit f'(x) = limₕ→0 [f(x+h) − f(x)] / h. In Edexcel papers, you may be asked to prove from first principles that the derivative of x² is 2x or that the derivative of sin x is cos x.
导数由极限 f'(x) = limₕ→0 [f(x+h) − f(x)] / h 定义。在爱德思试卷中，可能要求你从第一原理证明 x² 的导数为 2x，或 sin x 的导数为 cos x。
For f(x) = x², expand (x+h)² = x² + 2xh + h², subtract f(x), divide by h, and then let h tend to 0. Writing each algebraic step clearly earns method marks even if the final simplification is rushed.
对于 f(x) = x²，展开 (x+h)² = x² + 2xh + h²，减去 f(x)，再除以 h，最后令 h 趋于 0。清晰写出每一步代数变形，即使最后化简仓促也能获得方法分。
3. Core Differentiation Rules | 核心微分法则
The power rule states d/dx (xⁿ) = n xⁿ⁻¹ for any real constant n. Linear combinations follow term by term: d/dx [af(x)+bg(x)] = a f'(x) + b g'(x).
幂函数法则为 d/dx (xⁿ) = n xⁿ⁻¹，其中 n 为任意实常数。线性组合可逐项求导：d/dx [af(x)+bg(x)] = a f'(x) + b g'(x)。
The product rule and quotient rule are essential when functions are multiplied or divided. For u = u(x) and v = v(x), d/dx (uv) = u’ v + u v’, and d/dx (u/v) = (u’ v − u v’) / v².
当函数相乘或相除时，乘积法则和商法则是必不可少的。设 u = u(x)、v = v(x)，则 d/dx (uv) = u’ v + u v’，且 d/dx (u/v) = (u’ v − u v’) / v²。
4. Chain Rule and Connected Rates | 链式法则与相关变化率
The chain rule d/dx f(g(x)) = f'(g(x)) g'(x) is perhaps the most heavily tested differentiation technique. It is used for powers of brackets, trigonometric functions of linear angles, and connected rates of change.
链式法则 d/dx f(g(x)) = f'(g(x)) g'(x) 可能是考查最频繁的微分技巧。它用于括号幂、线性角度的三角函数，以及相关变化率问题。
For connected rates, if the radius r of a sphere increases at 3 cm s⁻¹ and volume V = 4/3 π r³, then dV/dt = dV/dr × dr/dt = 4π r² × 3. Substitute the given radius to find the rate of volume change.
对于相关变化率，如果球体半径 r 以 3 cm s⁻¹ 增加，且体积 V = 4/3 π r³，则 dV/dt = dV/dr × dr/dt = 4π r² × 3。代入给定半径即可求出体积变化率。
5. Applications of Differentiation | 微分的应用
Differentiation is used to find equations of tangents and normals. At a point (a, f(a)), the tangent has gradient f'(a), so its equation is y − f(a) = f'(a) (x − a). The normal has gradient −1 / f'(a).
微分用于求切线和法线方程。在点 (a, f(a)) 处，切线斜率为 f'(a)，因此切线方程为 y − f(a) = f'(a) (x − a)。法线斜率为 −1 / f'(a)。
Stationary points occur where f'(x) = 0. Use the second derivative or a sign table to classify them: f”(x) > 0 gives a local minimum, f”(x) < 0 gives a local maximum, and f”(x) = 0 requires further investigation.
驻点出现在 f'(x) = 0 处。使用二阶导数或符号表进行分类：f”(x) > 0 为局部极小值，f”(x) < 0 为局部极大值，f”(x) = 0 需要进一步检验。
6. Integration as the Reverse Process | 积分作为逆过程
Integration reverses differentiation. The indefinite integral ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c, provided n ≠ −1. Remember the constant of integration c because an indefinite integral represents a family of curves.
积分是微分的逆运算。不定积分 ∫ xⁿ dx = xⁿ⁺¹ / (n+1) + c，其中 n ≠ −1。务必记得加积分常数 c，因为不定积分表示一族曲线。
Definite integrals ∫ₐᵇ f(x) dx are evaluated as F(b) − F(a), where F is any antiderivative of f. The constant c cancels, so it is not needed in definite integration.
定积分 ∫ₐᵇ f(x) dx 的计算方法为 F(b) − F(a)，其中 F 是 f 的任一原函数。常数 c 会抵消，因此定积分中不需要加 c。
7. Integration Techniques | 积分技巧
For linear functions, ∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + c. Reverse chain rule works when a derivative of an inner function is present: ∫ f'(g(x)) g'(x) dx = f(g(x)) + c.
对于线性函数，∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + c。当被积函数中含有内层函数的导数时，可使用逆链式法则：∫ f'(g(x)) g'(x) dx = f(g(x)) + c。
Integration by substitution is required for more complex integrals. For example, to integrate ∫ x(2x²+1)⁵ dx, set u = 2x²+1, then du/dx = 4x, so x dx = du/4, and the integral becomes ¼ ∫ u⁵ du.
更复杂的积分需要换元积分法。例如，要计算 ∫ x(2x²+1)⁵ dx，设 u = 2x²+1，则 du/dx = 4x，因此 x dx = du/4，积分变为 ¼ ∫ u⁵ du。
8. Definite Integration and Area | 定积分与面积
The area between a curve y = f(x), the x-axis, and lines x = a and x = b is given by ∫ₐᵇ f(x) dx, provided f(x) ≥ 0 on [a, b]. If the curve lies below the axis, the integral is negative, so take the absolute value for area.
曲线 y = f(x)、x 轴及直线 x = a、x = b 之间的面积为 ∫ₐᵇ f(x) dx，前提是在 [a, b] 上 f(x) ≥ 0。如果曲线位于 x 轴下方，积分为负，计算面积时应取绝对值。
When a region is bounded by two curves, find the area as ∫ₐᵇ [f(x) − g(x)] dx, where f(x) ≥ g(x) on the interval. Always find the intersection points to determine a and b.
当区域由两条曲线围成时，面积为 ∫ₐᵇ [f(x) − g(x)] dx，其中在区间上 f(x) ≥ g(x)。务必先求出交点以确定 a 和 b。
9. Trigonometric Functions | 三角函数微分与积分
Key derivatives include d/dx (sin x) = cos x, d/dx (cos x) = −sin x, and d/dx (tan x) = sec² x. For linear angles, use the chain rule: d/dx (sin 2x) = 2 cos 2x.
关键导数包括 d/dx (sin x) = cos x、d/dx (cos x) = −sin x、d/dx (tan x) = sec² x。对于线性角度，使用链式法则：d/dx (sin 2x) = 2 cos 2x。
For integration, ∫ cos x dx = sin x + c, ∫ sin x dx = −cos x + c, and ∫ sec² x dx = tan x + c. The double-angle identities are often needed to integrate sin² x or cos² x.
积分方面，∫ cos x dx = sin x + c，∫ sin x dx = −cos x + c，∫ sec² x dx = tan x + c。积分 sin² x 或 cos² x 时经常需要用到二倍角公式。
10. Exponentials and Logarithms | 指数与对数函数的微积分
The exponential function eˣ is unique: d/dx (eˣ) = eˣ and ∫ eˣ dx = eˣ + c. For eᵏˣ, the chain rule gives d/dx (eᵏˣ) = k eᵏˣ and ∫ eᵏˣ dx = (1/k) eᵏˣ + c.
指数函数 eˣ 具有独特性质：d/dx (eˣ) = eˣ，∫ eˣ dx = eˣ + c。对于 eᵏˣ，链式法则给出 d/dx (eᵏˣ) = k eᵏˣ，∫ eᵏˣ dx = (1/k) eᵏˣ + c。
The natural logarithm function has derivative d/dx (ln x) = 1/x for x > 0. Its integral is ∫ 1/x dx = ln |x| + c. The absolute value ensures the logarithm is defined for negative x as well.
自然对数函数的导数为 d/dx (ln x) = 1/x（x > 0）。其积分为 ∫ 1/x dx = ln |x| + c。绝对值保证了当 x 为负时对数也有定义。
11. Numerical Methods and Exam Skills | 数值方法与考试技巧
The trapezium rule approximates ∫ₐᵇ f(x) dx using h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ], where h = (b − a)/n. It is particularly useful when the integral cannot be found analytically.
梯形法则用公式 h/2 [y₀ + 2(y₁ + y₂ + … + yₙ₋₁) + yₙ] 近似计算 ∫ₐᵇ f(x) dx，其中 h = (b − a)/n。当积分无法解析求出时，该方法尤其有用。
In exam questions, underline key instructions and units. When solving optimisation problems, write a clear equation for the quantity to be maximised or minimised, then differentiate, set the derivative to zero, and justify the nature of the stationary point.
考试答题时，划出关键指令和单位。解决最优化问题时，先明确写出要最大化或最小化的量的表达式，然后求导、令导数为零，并说明驻点的性质。
12. Summary and Revision Checklist | 总结与复习清单
Before the exam, ensure you can differentiate all standard functions, reverse the process to integrate, use the chain rule both ways, find stationary points, calculate areas, and interpret rates of change. Practise past paper questions under timed conditions.
考试前，确保你能对所有标准函数求导，并能逆向进行积分；能双向使用链式法则；能求驻点、计算面积并解释变化率。在限时条件下练习历年真题。
A useful checklist includes: first-principles proof, product and quotient rules, tangents and normals, optimisation, definite vs indefinite integrals, area between curves, trigonometric integrals, exponential/log derivatives, and trapezium rule.
一份有用的清单包括：第一原理证明、乘积与商法则、切线与法线、最优化、定积分与不定积分的区别、曲线间面积、三角积分、指数/对数导数以及梯形法则。
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