📚 IGCSE WJEC Maths: Calculus Basics – Key Points Review | IGCSE WJEC 数学：微积分基础 考点精讲
Calculus is one of the most powerful tools in mathematics, allowing us to analyse change, motion, and accumulation. For the IGCSE WJEC Mathematics specification, calculus appears as an introduction to differentiation and integration, focusing on polynomial functions, the gradient of curves, stationary points, and finding areas under curves. Mastering these basics not only secures high marks in the exam but also builds a strong foundation for further study in A-Level or IB mathematics. This article provides a clear, bilingual breakdown of every essential concept, worked examples, and common pitfalls.
微积分是数学中最强大的工具之一，使我们能够分析变化、运动和累积。在 IGCSE WJEC 数学大纲中，微积分作为导数和积分的入门，重点在于多项式函数、曲线梯度、驻点以及曲线下方面积的计算。掌握这些基础知识不仅能帮助你在考试中取得高分，也为 A-Level 或 IB 数学的深造打下坚实基础。本文以中英对照的方式，详细梳理每个核心概念、典型例题和常见失分点。
1. What is Differentiation? | 什么是微分？
Differentiation is the process of finding the derivative of a function. The derivative measures how a function’s output changes as its input changes; in geometrical terms, it gives the gradient of the tangent line to a curve at any point.
微分是求函数导数的过程。导数衡量的是函数输出如何随输入变化而变化；从几何角度看，它给出了曲线上任意一点切线的斜率。
If we have a function y = f(x), the derivative is often written as f ‘(x) or dy/dx. This notation means ‘the rate of change of y with respect to x’. For example, if y represents distance and x represents time, dy/dx represents speed.
如果有一个函数 y = f(x)，其导数通常记作 f ‘(x) 或 dy/dx。这一符号表示’y 关于 x 的变化率’。例如，若 y 表示距离，x 表示时间，则 dy/dx 就表示速度。
The basic idea can be approached through limits, but at IGCSE level you are expected to apply the differentiation rules directly, without formal limit proofs. The key is to understand that differentiating a function produces another function that gives the steepness of the original graph at every point.
基本思想可以通过极限来理解，但在 IGCSE 阶段，你只需直接应用求导法则，无需进行正式的极限证明。关键是理解：对一个函数求导会得到另一个函数，该函数给出原图形在每一点的倾斜程度。
2. The Power Rule for Differentiation | 幂函数求导法则
The most important rule for IGCSE is the power rule. For any term of the form axⁿ, where a is a constant and n is a real number, the derivative is n × a × xⁿ⁻¹. In words: multiply by the power, then reduce the power by 1.
IGCSE 阶段最重要的法则是幂法则。对于形如 axⁿ 的项，其中 a 是常数，n 是实数，其导数为 n × a × xⁿ⁻¹。简而言之：乘以指数，再将指数减 1。
The constant term disappears when differentiated because the derivative of a constant is 0. This is consistent with the power rule if we write a constant as c × x⁰ (since x⁰ = 1), giving derivative 0 × c × x⁻¹ = 0.
常数项求导后为零，因为常数的导数为 0。如果我们将常数写成 c × x⁰（x⁰ = 1），则根据幂法则求导得到 0 × c × x⁻¹ = 0，这与法则一致。
For example, if y = 5x³, then dy/dx = 3 × 5 × x² = 15x². If y = 4x, we have x¹, so dy/dx = 1 × 4 × x⁰ = 4.
例如，若 y = 5x³，则 dy/dx = 3 × 5 × x² = 15x²。若 y = 4x，这相当于 x¹，因此 dy/dx = 1 × 4 × x⁰ = 4。
If f(x) = xⁿ, then f ‘(x) = nxⁿ⁻¹
3. Differentiating Polynomials Step by Step | 多项式求导步骤
A polynomial is a sum of terms like axⁿ. To differentiate a polynomial, apply the power rule to each term independently, then add the results. This is called term-by-term differentiation.
多项式是形如 axⁿ 的各项之和。要对多项式求导，只需对每一项分别应用幂法则，再将结果相加。这称为逐项求导。
Steps: 1) Identify each term; 2) For each term, multiply the coefficient by the power and write down; 3) Subtract 1 from the power to get the new exponent; 4) Add up the derived terms; 5) Omit any constant terms (they become 0).
步骤：1) 识别每一单项；2) 对每一项，系数乘以指数并将结果写下；3) 指数减 1 得到新的幂次；4) 将求导后的各项相加；5) 去掉所有常数项（它们变为 0）。
Example: Differentiate y = 2x³ – 5x² + 3x – 7. Derivative: dy/dx = (3×2)x² – (2×5)x¹ + (1×3)x⁰ – 0 = 6x² – 10x + 3.
例题：对 y = 2x³ – 5x² + 3x – 7 求导。导数：dy/dx = (3×2)x² – (2×5)x¹ + (1×3)x⁰ – 0 = 6x² – 10x + 3。
Be careful with minus signs: when a term has a negative coefficient, the derivative keeps that negative sign after multiplication. Also remember that x¹ becomes x⁰, which equals 1.
注意负号：当一项系数为负时，求导后经乘法仍需保留该负号。同时记住 x¹ 变为 x⁰，即等于 1。
4. Finding the Gradient of a Tangent | 求切线斜率
Once you have the derivative function f ‘(x), you can find the gradient of the curve at any specific point by substituting the x-coordinate of that point into the derivative.
一旦求得导函数 f ‘(x)，只需将某点的 x 坐标代入导数，即可求出曲线在该点的切线斜率。
For example, given y = x² + 2x, find the gradient of the tangent at x = 3. First, find dy/dx = 2x + 2. Then substitute x = 3: gradient = 2(3) + 2 = 8. This tells us the curve is quite steep at that point.
例如，已知 y = x² + 2x，求在 x = 3 处的切线斜率。首先求导得 dy/dx = 2x + 2。再代入 x = 3：斜率 = 2(3) + 2 = 8。这表明曲线在该点处非常陡峭。
This technique is fundamental for understanding the shape of a graph and for solving problems involving parallel or perpendicular lines to the tangent.
这一技巧对于理解图形形状以及解决与切线平行或垂直的直线问题至关重要。
5. Equation of Tangent and Normal | 切线与法线方程
The tangent line at a point on a curve touches the curve at that point and has the same gradient as the curve there. The normal line is perpendicular to the tangent. To find the equation of a tangent or normal, you need a point on the curve and the gradient at that point.
曲线上某点的切线在该点与曲线相切，且与该点处的曲线斜率相同。法线则垂直于切线。要求切线或法线的方程，需要曲线上的一点以及该点处的斜率。
For a point (x₁, y₁) and gradient m, the equation of the line is y – y₁ = m(x – x₁). For the normal, the gradient is the negative reciprocal of the tangent’s gradient: mₙₒᵣₘₐₗ = -1/mₜₐₙ₉ₑₙₜ.
对于点 (x₁, y₁) 和斜率 m，直线方程为 y – y₁ = m(x – x₁)。对于法线，其斜率为切线斜率的负倒数：mₙₒᵣₘₐₗ = -1/mₜₐₙ₉ₑₙₜ。
Example: Find the tangent and normal to y = x² – 4x at x = 5. First, y-coordinate: y = 5² – 4(5) = 5. Derivative dy/dx = 2x – 4; at x = 5, m = 6. Tangent equation: y – 5 = 6(x – 5) → y = 6x – 25. Normal gradient = -1/6; normal equation: y – 5 = -1/6 (x – 5) → y = -1/6 x + 35/6.
例题：求曲线 y = x² – 4x 在 x = 5 处的切线与法线。首先求 y 坐标：y = 5² – 4(5) = 5。导数 dy/dx = 2x – 4；在 x = 5 时，m = 6。切线方程：y – 5 = 6(x – 5) → y = 6x – 25。法线斜率 = -1/6；法线方程：y – 5 = -1/6 (x – 5) → y = -1/6 x + 35/6。
Always remember to calculate the y-coordinate from the original function before using the point in the line equation.
务必先由原函数计算出 y 坐标，再代入直线方程中使用该点。
6. The Second Derivative | 二阶导数
If you differentiate a function twice, you obtain the second derivative, denoted by f ”(x) or d²y/dx². This tells you how the gradient itself is changing. While the first derivative gives the slope, the second derivative indicates the concavity of the curve.
对函数求导两次，就得到二阶导数，记为 f ”(x) 或 d²y/dx²。它表示梯度本身的变化情况。一阶导数给出斜率，二阶导数则指示曲线的凹凸性。
To find the second derivative, simply differentiate the first derivative using the same power rule. For instance, if y = 3x⁴, first derivative dy/dx = 12x³, then second derivative d²y/dx² = 36x².
求二阶导数只需对一阶导数再次使用幂法则。例如，若 y = 3x⁴，一阶导数 dy/dx = 12x³，接着二阶导数 d²y/dx² = 36x²。
At IGCSE level, the second derivative is mainly tested in the context of determining the nature of stationary points (maximum or minimum) when the first derivative is zero. A positive second derivative indicates a minimum point, while a negative second derivative indicates a maximum point.
在 IGCSE 阶段，二阶导数主要用来判断当一阶导数为零时驻点的性质（极大值或极小值）。二阶导数为正表明是极小值点，二阶导数为负表明是极大值点。
If f ‘(x) = 0 and f ”(x) > 0 → local minimum; f ”(x) < 0 → local maximum
7. What is Integration? | 什么是积分？
Integration is the reverse process of differentiation. If you are given a derivative, integrating it gives back the original function (plus an unknown constant). For this reason, integration is often called anti-differentiation.
积分是微分的逆运算。如果已知一个导数，对其进行积分就能还原出原函数（外加一个未知常数）。因此，积分常被称为反导数。
In IGCSE WJEC, you will meet indefinite integrals, which produce a family of functions, and definite integrals, which compute a numerical value representing the area under a curve between two x-values.
在 IGCSE WJEC 中，你会遇到不定积分，它得到一族函数；以及定积分，它计算出一个数值，表示在两个 x 值之间曲线下方的面积。
The integral sign is ∫, and the differential dx indicates the variable of integration. For example, ∫ 2x dx = x² + C, where C is the constant of integration. Notice that differentiating x² + C gives 2x back.
积分符号为 ∫，微分 dx 表示积分变量。例如，∫ 2x dx = x² + C，其中 C 为积分常数。注意，对 x² + C 求导确实还原为 2x。
8. Indefinite Integrals and the Constant of Integration | 不定积分与积分常数
The power rule for integration is the inverse of differentiation: for any term axⁿ (n ≠ -1), the integral is (a/(n+1))xⁿ⁺¹ + C. In words: increase the power by 1, then divide by the new power, and finally add the constant of integration.
积分的幂法则正是求导的逆运算：对于任意项 axⁿ（n ≠ -1），其积分为 (a/(n+1))xⁿ⁺¹ + C。口诀：指数加 1，除以新指数，最后加上积分常数。
The constant C is crucial because when you differentiate, any constant term vanishes. Integration must account for all possible original functions that differ by a constant. Forgetting + C is a very common mistake in exams.
积分常数 C 至关重要，因为在求导过程中任何常数项都会消失。积分必须考虑到所有相差一个常数的原函数。忘记加 + C 是考试中最常见的错误。
Example: Find ∫ (3x² + 4x – 1) dx. Integrate term by term: 3x² → 3 × (x³/3) = x³; 4x → 4 × (x²/2) = 2x²; -1 → -1 × (x¹/1) = -x. So the result is x³ + 2x² – x + C.
例题：求 ∫ (3x² + 4x – 1) dx。逐项积分：3x² → 3 × (x³/3) = x³；4x → 4 × (x²/2) = 2x²；-1 → -1 × (x¹/1) = -x。因此结果为 x³ + 2x² – x + C。
If additional information is given, such as a point on the original curve, you can solve for C. For example, if f ‘(x) = 2x and f(1) = 3, then f(x) = x² + C. Substituting (1,3) gives 3 = 1 + C, so C = 2, hence f(x) = x² + 2.
若题目给出了额外信息，比如原曲线上某点的坐标，你就可以解出 C。例如，已知 f ‘(x) = 2x 且 f(1) = 3，则 f(x) = x² + C。代入 (1,3) 得 3 = 1 + C，因此 C = 2，从而 f(x) = x² + 2。
9. Definite Integrals and Calculating Area | 定积分和面积计算
A definite integral has upper and lower limits, for example ∫ₐᵇ f(x) dx. Evaluating it gives a number, not a function. To compute it, first find the indefinite integral F(x) (without + C), then calculate F(b) – F(a).
定积分带有上限和下限，例如 ∫ₐᵇ f(x) dx。计算出的结果是一个数值，而不是一个函数。计算步骤：先求出不定积分 F(x)（不加 + C），再计算 F(b) – F(a)。
The value of a definite integral equals the signed area between the curve and the x-axis from x = a to x = b. Areas above the x-axis are positive, areas below are negative. Always sketch a diagram to understand the region.
定积分的值等于曲线与 x 轴之间从 x = a 到 x = b 的带有符号的面积。x 轴上方的面积为正，下方的面积为负。一定要画出示意图来理解所求区域。
Example: Evaluate ∫₁³ (2x) dx. First, integrate: ∫ 2x dx = x². Then substitute limits: [x²]₁³ = 3² – 1² = 9 – 1 = 8. This represents the area under the line y = 2x from x = 1 to x = 3.
例题：计算 ∫₁³ (2x) dx。首先积分：∫ 2x dx = x²。然后代入上下限：[x²]₁³ = 3² – 1² = 9 – 1 = 8。这表示直线 y = 2x 下 x = 1 到 x = 3 之间的面积。
If the curve crosses the x-axis within the interval, you must split the integral into separate parts where the function is positive and negative, taking the absolute value of each part to find the total area. This is a typical WJEC exam question.
如果曲线在区间内穿过 x 轴，则必须将积分拆分成函数为正和为负的若干部分，对各部分取绝对值再相加，从而求得总面积。这是 WJEC 考试中的典型题目。
Area = ∫ₐᵇ |f(x)| dx = |∫ₐᶜ f(x) dx| + |∫ᶜᵇ f(x) dx| if f(c)=0
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
- Forgetting the constant of integration in indefinite integrals — always write ‘+ C’ — 忘记不定积分中的积分常数：务必写上 ‘+ C’
- Misapplying the power rule: when differentiating, remember to multiply by the power first, then subtract 1, not the other way round — 幂法则应用错误：求导时要先乘以指数，然后指数减 1，不要颠倒顺序
- Sign errors when integrating negative terms — check that dividing by the new power preserves the sign — 积分负项时出现符号错误：检查除以新指数后符号是否保持不变
- Confusing the derivative of a constant: the derivative of any constant is 0, but the integral of 0 is a constant — 混淆常数的导数：任何常数的导数为 0，但 0 的积分却是一个常数
- Forgetting to find the y-coordinate of a point when writing tangent/normal equations — 书写切线/法线方程时忘记求点的 y 坐标
- Not simplifying the function before differentiating or integrating (e.g., write 1/x² as x⁻² to use the power rule) — 在求导或积分前未化简函数（例如将 1/x² 写成 x⁻² 以便使用幂法则）
Practice is essential. Work through past paper questions under timed conditions. When tackling area problems, always draw the graph first, identify intervals, and consider whether the area is above or below the axis.
练习至关重要。在限时条件下完成历年真题。在处理面积问题时，务必先绘出图形，确定区间，并考虑面积是在 x 轴上方还是下方。
Finally, present your working clearly. Even if the final answer is wrong, method marks are often awarded for correctly finding derivatives, setting up integrals, or applying limits. Write each step separately and label them.
最后，解题步骤要清晰明了。即使最终答案出错，只要导数、积分表达式或上下限代入等步骤正确，通常都能得到步骤分。一步一步写出并标注清楚。
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