📚 Core Calculus Techniques for Edexcel A-Level Pure Mathematics | 爱德思A-Level纯数学核心微积分方法
Calculus lies at the heart of Edexcel A-Level Pure Mathematics. This article brings together the key differentiation and integration techniques, common exam traps, and efficient strategies. Work through each section in order, then test yourself with past-paper questions.
微积分是爱德思A-Level纯数学的核心。本文汇总关键微分与积分技巧、常见考试陷阱和高效解题策略。按顺序学习每节内容，再用历年真题进行自测。
1. Differentiation from First Principles | 从第一性原理求导
The gradient of a chord between (x, f(x)) and (x+h, f(x+h)) gives the average rate of change. As h → 0, this chord approaches the tangent, so the derivative is defined by the limit below.
连接点 (x, f(x)) 与 (x+h, f(x+h)) 的弦的斜率给出平均变化率。当 h → 0 时，这条弦趋近于切线，因此导数由下面的极限定义。
f'(x) = lim(h → 0) [f(x + h) − f(x)] ÷ h
For f(x) = x², expand (x+h)² − x² = 2xh + h². Divide by h to get 2x + h, then let h → 0. The derivative is therefore 2x. Edexcel often tests this proof in Pure Paper 1.
对于 f(x) = x²，展开 (x+h)² − x² = 2xh + h²。除以 h 得到 2x + h，再令 h → 0。因此导数为 2x。爱德思常在纯数卷一考查这个证明。
Always write the limit expression first, then simplify before substituting h = 0. If you substitute h = 0 too early, you may obtain the meaningless form 0 ÷ 0.
一定要先写出极限表达式，再化简，最后才代入 h = 0。如果过早代入 h = 0，可能会得到无意义的 0 ÷ 0 形式。
2. Standard Derivatives and the Chain Rule | 标准导数与链式法则
Memorise the standard results: d/dx(xⁿ) = n xⁿ⁻¹, d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, d/dx(sin x) = cos x, d/dx(cos x) = −sin x, d/dx(tan x) = sec² x.
熟记标准结果：d/dx(xⁿ) = n xⁿ⁻¹，d/dx(eˣ) = eˣ，d/dx(ln x) = 1/x，d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(tan x) = sec² x。
If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). In words: differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function.
若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。用语言描述就是：先对外层函数求导，保持内层函数不变，再乘以内层函数的导数。
dy/dx = f'(g(x)) × g'(x)
Example: if y = (3x² + 2)⁵, let u = 3x² + 2. Then dy/du = 5u⁴ and du/dx = 6x, so dy/dx = 30x(3x² + 2)⁴.
例子：若 y = (3x² + 2)⁵，设 u = 3x² + 2。则 dy/du = 5u⁴，du/dx = 6x，所以 dy/dx = 30x(3x² + 2)⁴。
3. Product and Quotient Rules | 乘积法则与商法则
The product rule is used when y = u v, where both u and v are functions of x. The derivative is u times the derivative of v plus v times the derivative of u.
当 y = u v 且 u 和 v 都是 x 的函数时使用乘积法则。导数为 u 乘以 v 的导数加上 v 乘以 u 的导数。
dy/dx = u dv/dx + v du/dx
For y = x² eˣ, choose u = x² and v = eˣ. Then dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2). Factorising is useful when finding stationary points later.
对于 y = x² eˣ，取 u = x²，v = eˣ。则 dy/dx = x² eˣ + 2x eˣ = x eˣ(x + 2)。在后续求驻点时，因式分解会很有用。
The quotient rule is used when y = u/v. The formula has a minus sign, so get the numerator order right: v du/dx − u dv/dx, not the reverse.
当 y = u/v 时使用商法则。公式中有减号，因此分子顺序要写对：v du/dx − u dv/dx，而不是反过来。
dy/dx = (v du/dx − u dv/dx) ÷ v²
Example: if y = sin x / x, then dy/dx = [x cos x − sin x] ÷ x². It is often better to leave the denominator as v² rather than expanding it.
例子：若 y = sin x / x，则 dy/dx = [x cos x − sin x] ÷ x²。通常最好将分母保留为 v²，而不要展开。
4. Parametric Differentiation | 参数微分
When x = f(t) and y = g(t), the derivative dy/dx is not simply dy/dt. You must divide dy/dt by dx/dt, provided dx/dt ≠ 0.
当 x = f(t)、y = g(t) 时，dy/dx 并不只是 dy/dt。你必须用 dy/dt 除以 dx/dt，并且要求 dx/dt ≠ 0。
dy/dx = (dy/dt) ÷ (dx/dt)
Example: if x = t² + 1 and y = 2t³ − t, then dx/dt = 2t and dy/dt = 6t² − 1. Hence dy/dx = (6t² − 1) ÷ 2t.
例子：若 x = t² + 1，y = 2t³ − t，则 dx/dt = 2t，dy/dt = 6t² − 1。因此 dy/dx = (6t² − 1) ÷ 2t。
For the second derivative, use d²y/dx² = d/dt(dy/dx) ÷ dx/dt. Do not confuse d²y/dx² with d²y/dt² divided by d²x/dt².
对于二阶导数，使用 d²y/dx² = d/dt(dy/dx) ÷ dx/dt。不要把 d²y/dx² 与 d²y/dt² 除以 d²x/dt² 混淆。
d²y/dx² = d/dt(dy/dx) ÷ dx/dt
5. Implicit Differentiation | 隐函数微分
When a curve is given as an equation in x and y, differentiate both sides with respect to x. Treat y as a function of x, so every y-term gains a dy/dx factor by the chain rule.
当曲线由 x 和 y 的方程给出时，对两边关于 x 求导。将 y 视为 x 的函数，因此每个含 y 的项根据链式法则都要乘上 dy/dx。
For x² + y² = 25, differentiating gives 2x + 2y(dy/dx) = 0. Rearranging yields dy/dx = −x/y. At the point (3, 4), the gradient is −3/4.
对于 x² + y² = 25，求导得 2x + 2y(dy/dx) = 0。整理得 dy/dx = −x/y。在点 (3, 4) 处，斜率为 −3/4。
When differentiating products such as xy², you must use the product rule together with implicit differentiation. This is a common source of sign and missing-factor errors.
当对 xy² 这类乘积求导时，必须同时使用乘积法则和隐函数微分。这是常见的符号错误和漏乘因子错误来源。
6. Applications of Differentiation: Tangents, Normals, Stationary Points | 微分应用：切线、法线、驻点
Once dy/dx is known at a point, the tangent equation is y − y₁ = m(x − x₁), where m = dy/dx. The normal gradient is −1/m.
一旦已知某点的 dy/dx，切线方程为 y − y₁ = m(x − x₁)，其中 m = dy/dx。法线斜率为 −1/m。
Tangent: y − y₁ = m(x − x₁); Normal: y − y₁ = (−1/m)(x − x₁)
Stationary points occur where dy/dx = 0. Classify them using the second derivative: d²y/dx² > 0 gives a local minimum, d²y/dx² < 0 gives a local maximum, and d²y/dx² = 0 requires a sign table or further investigation.
驻点出现在 dy/dx = 0 处。使用二阶导数进行分类：d²y/dx² > 0 为局部极小值，d²y/dx² < 0 为局部极大值，d²y/dx² = 0 时需要用符号表或进一步分析。
Example: for y = x³ − 3x, dy/dx = 3x² − 3 = 3(x² − 1). Stationary points are x = 1 and x = −1. Second derivative is 6x, so x = 1 is a minimum and x = −1 is a maximum.
例子：对于 y = x³ − 3x，dy/dx = 3x² − 3 = 3(x² − 1)。驻点为 x = 1 和 x = −1。二阶导数为 6x，所以 x = 1 是极小值点，x = −1 是极大值点。
7. Integration as Reverse Differentiation | 积分作为微分的逆运算
Indefinite integration reverses differentiation. The general results are listed below; the constant of integration C must always be included for indefinite integrals.
不定积分是微分的逆运算。一般结果如下；不定积分必须始终加上积分常数 C。
∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C for n ≠ −1; ∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C; ∫ sin x dx = −cos x + C; ∫ cos x dx = sin x + C
Remember that the derivative of ln(x) is 1/x, so the integral of 1/x is ln|x|, not ln x. The absolute value avoids issues with negative x.
记住 ln(x) 的导数是 1/x，所以 1/x 的积分是 ln|x|，不是 ln x。绝对值可避免 x 为负时的问题。
For definite integrals, evaluate the antiderivative at the upper limit and subtract the value at the lower limit. No constant of integration is needed in this case.
对于定积分，先求原函数，再代入上限求值并减去下限求值。此时不需要积分常数。
8. Integration by Substitution | 换元积分法
Substitution reverses the chain rule. Choose u = g(x), then replace dx by du / g'(x). The aim is to convert the integral into a standard form in u.
换元积分法逆转链式法则。选择 u = g(x)，然后用 du / g'(x) 替换 dx。目标是使积分变成关于 u 的标准形式。
Example: for ∫ x√(x² + 1) dx, let u = x² + 1, so du/dx = 2x and x dx = (1/2) du. The integral becomes (1/2) ∫ u^½ du = (1/2) × (2/3) u^3/2 + C = (1/3)(x² + 1)^3/2 + C.
例子：对于 ∫ x√(x² + 1) dx，设 u = x² + 1，则 du/dx = 2x，x dx = (1/2) du。积分变为 (1/2) ∫ u^½ du = (1/2) × (2/3) u^3/2 + C = (1/3)(x² + 1)^3/2 + C。
For definite integrals, change the limits at the moment you substitute. If the original limits are x = a and x = b, the new limits are u = g(a) and u = g(b). Do not substitute back and then use the old limits.
对于定积分，在换元的同时就要更换上下限。如果原上下限是 x = a 和 x = b，新上下限就是 u = g(a) 和 u = g(b)。不要先换回原变量再使用旧上下限。
9. Integration by Parts | 分部积分法
Integration by parts reverses the product rule. The formula is given below. Choose u and dv so that ∫ v du is easier than the original integral.
分部积分法逆转乘积法则。公式如下。选择 u 和 dv 时，应使 ∫ v du 比原积分更容易计算。
∫ u dv = u v − ∫ v du
A good order for choosing u is ‘LATE’: Logarithmic, Algebraic, Trigonometric, Exponential. For example, in ∫ x eˣ dx choose u = x because algebraic functions come before exponential ones.
选择 u 的一个好顺序是 ‘LATE’：对数函数、代数函数、三角函数、指数函数。例如，在 ∫ x eˣ dx 中选择 u = x，因为代数函数排在指数函数之前。
For ∫ x eˣ dx, set u = x and dv/dx = eˣ. Then du/dx = 1 and v = eˣ, so the integral becomes x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.
对于 ∫ x eˣ dx，设 u = x，dv/dx = eˣ。则 du/dx = 1，v = eˣ，因此积分变为 x eˣ − ∫ eˣ dx = x eˣ − eˣ + C。
When using parts twice, as with ∫ x² eˣ dx, keep the same choice of u and dv in the second application, otherwise the working can easily unravel.
当需要两次使用分部积分时，例如 ∫ x² eˣ dx，第二次应用时 u 和 dv 的选择要保持一致，否则过程很容易混乱。
10. Definite Integration and Area | 定积分与面积
A definite integral gives signed area, meaning that area below the x-axis is counted as negative. To find the true total area between a curve and the x-axis, split the interval at each root.
定积分给出带符号的面积，即 x 轴下方的面积计为负值。要求曲线与 x 轴之间的真实总面积，必须在每个根处拆分区间。
Example: for y = x(x − 2) between x = 0 and x = 3, the curve crosses the x-axis at x = 2. Compute |∫₀² y dx| + |∫₂³ y dx|, not simply ∫₀³ y dx.
例子：对于 y = x(x − 2)，在 x = 0 到 x = 3 之间，曲线在 x = 2 处穿过 x 轴。应计算 |∫₀² y dx| + |∫₂³ y dx|，而不是简单地计算 ∫₀³ y dx。
The area between two curves y = f(x) and y = g(x) is found by integrating the top curve minus the bottom curve over the given interval. Always identify which curve is higher before integrating.
两条曲线 y = f(x) 与 y = g(x) 之间的面积，通过在被积区间内对“上方曲线减去下方曲线”进行积分求得。积分前一定要先判断哪条曲线在上方。
Area = ∫ [top curve − bottom curve] dx
11. Differential Equations | 微分方程
A separable differential equation can be written with all y terms on one side and all x terms on the other. Integrate both sides and include one constant of integration.
可分离变量的微分方程可以将所有含 y 的项放在一边，所有含 x 的项放在另一边。两边积分，并只加一个积分常数。
dy/dx = ky → ∫ (1/y) dy = ∫ k dx → ln|y| = kx + C → y = A eᵏˣ
The exponential model y = A eᵏˣ describes exponential growth when k > 0 and exponential decay when k < 0. Use an initial condition to find A, then any further condition to find k.
指数模型 y = A eᵏˣ 在 k > 0 时描述指数增长，在 k < 0 时描述指数衰减。使用初始条件求 A，再用另一条件求 k。
In Edexcel questions, the differential equation may be given in context, such as temperature change or population growth. Translate the rate phrase carefully: ‘rate of change is proportional to the current amount’ means dy/dx = ky.
在爱德思考题中，微分方程可能以实际情境给出，例如温度变化或人口增长。仔细翻译速率语句：“变化率与当前数量成正比”意味着 dy/dx = ky。
12. Exam Technique and Common Errors | 考试技巧与常见错误
The most common errors in Edexcel calculus questions are: forgetting the constant in indefinite integration, reversing the quotient rule numerator, choosing a poor u in integration by parts, and not changing limits in substitution.
爱德思微积分题目中最常见的错误有：不定积分漏写常数、商法则分子顺序颠倒、分部积分中 u 选择不当、换元时未更换上下限。
Show clear working: write the formula you are using, state any substitution or choice of u, and simplify answers where the question asks for exact form. This gains method marks even if a numerical slip occurs.
展示清晰过程：写出所用公式，说明换元或 u 的选择，并在题目要求精确形式时化简结果。即使出现数字计算小错，也能获得方法分。
Before the exam, practise mixed differentiation and integration exercises under timed conditions. Review each error and ask whether it was a concept issue, a sign slip, or a missing condition.
考前应在限时条件下练习混合微积分题。复盘每个错误，判断是概念问题、符号疏漏还是漏用条件。
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