IGCSE WJEC Mathematics: Differentiation – Key Concepts and Revision Notes | IGCSE WJEC 数学：微分考点精讲

📚 IGCSE WJEC Mathematics: Differentiation – Key Concepts and Revision Notes | IGCSE WJEC 数学：微分考点精讲

Differentiation is a fundamental tool in calculus that allows us to find the rate at which a quantity changes. In the WJEC IGCSE Mathematics syllabus, you will learn how to differentiate simple polynomial functions, determine gradients of curves, find equations of tangents and normals, and identify maximum and minimum points. Mastering these skills is essential for solving real‑world problems involving optimisation and motion.

微分是微积分中的一个基本工具，能帮助我们求出一个量变化的快慢。在 WJEC IGCSE 数学考纲中，你将学习如何对简单的多项式函数求导、确定曲线的梯度、求切线和法线的方程，以及找出极大值和极小值点。掌握这些技能对解决涉及优化和运动的实际问题至关重要。

1. What Is Differentiation? | 什么是微分？

Differentiation is the process of finding the derivative of a function. The derivative at a particular point tells you the exact gradient (steepness) of the curve at that instant. It gives the instantaneous rate of change of y with respect to x.

微分是求一个函数导数的过程。某一点的导数告诉你曲线在该点的精确梯度（倾斜程度）。它给出了 y 相对于 x 的瞬时变化率。

Imagine a moving car: its speedometer shows the instantaneous speed, which is the derivative of distance with respect to time. In mathematics, if y = f(x), the derivative is often written as dy/dx or f'(x). Both notations represent the same concept – the slope of the tangent line to the curve y = f(x).

想象一辆行驶的汽车：车速表显示的是瞬时速度，它就是距离关于时间的导数。在数学中，如果 y = f(x)，导数通常写成 dy/dx 或 f'(x)。两种符号表示同一个概念——曲线 y = f(x) 切线的斜率。

2. Derivative Notation and Basic Rules | 导数符号与基本法则

The two most common notations you will see are dy/dx (Leibniz notation) and f'(x) (Lagrange notation). For example, if y = x², then dy/dx = 2x and f'(x) = 2x. Both mean exactly the same thing, so feel free to use whichever you prefer in your answers.

你最常见到的两种符号是 dy/dx（莱布尼茨记号）和 f'(x)（拉格朗日记号）。例如，若 y = x²，则 dy/dx = 2x，同时 f'(x) = 2x。它们含义完全相同，答题时选择你习惯的一种即可。

When we differentiate, we apply three key rules right away: the power rule, the constant multiple rule, and the sum/difference rule. Understanding these rules allows you to differentiate any polynomial term by term without memorising long tables.

求导时，我们会立即用到三条关键法则：幂函数法则、常数倍法则以及和差法则。掌握它们后，你就可以逐项对任意多项式求导，无需死记硬背长长的表格。

3. The Power Rule for xⁿ | xⁿ 的幂函数求导法则

The most important differentiation rule for IGCSE is the power rule: if y = xⁿ, then dy/dx = n xⁿ⁻¹. You simply bring the power down in front and reduce the original power by 1. This works for any real number n, but you will mainly use positive integer powers and perhaps simple fractions like ½.

IGCSE 最重要的求导法则就是幂法则：如果 y = xⁿ，那么 dy/dx = n xⁿ⁻¹。你只需把指数“搬”下来乘在前面，再把原来指数减 1。这适用于任何实数 n，但你主要会遇到正整数幂以及如 ½ 这样的简单分数。

For instance, differentiate y = x⁴: dy/dx = 4x³. Differentiate y = x⁻²: dy/dx = -2x⁻³. Even a constant raised to a power such as y = x⁰ (which equals 1) gives dy/dx = 0, because 0 × x⁻¹ = 0.

例如，对 y = x⁴ 求导：dy/dx = 4x³。对 y = x⁻² 求导：dy/dx = -2x⁻³。甚至常数次幂如 y = x⁰（等于 1）求导得 dy/dx = 0，因为 0 × x⁻¹ = 0。

Always rewrite roots as fractional powers before differentiating. For example, √x = x½, so its derivative is ½ x⁻½, which can be written as 1/(2√x).

求导前始终要把根式改写成分数指数形式。例如 √x = x½，它的导数就是 ½ x⁻½，也可以写成 1/(2√x)。

4. Differentiating Polynomials (Term by Term) | 多项式求导（逐项求导）

A polynomial is an expression made of terms like axⁿ added together. To differentiate a polynomial, simply apply the power rule to each term separately while keeping any constant multipliers attached. The derivative of a sum is the sum of the derivatives.

多项式是由形如 axⁿ 的项相加而成的表达式。要对多项式求导，只需分别对每一项使用幂法则，并保留项前的常系数。和的导数等于导数的和。

Example: Differentiate y = 5x³ – 2x² + 7x – 4. The derivative is dy/dx = 15x² – 4x + 7. The constant term (-4) has derivative 0. Remember: a constant on its own disappears when differentiated.

例题：求 y = 5x³ – 2x² + 7x – 4 的导数。结果是 dy/dx = 15x² – 4x + 7。常数项 (-4) 的导数为 0。请记住：单独的常数在求导后会消失。

The constant multiple rule tells you that if y = k u(x), then dy/dx = k × du/dx. This means you can differentiate the variable part first and then multiply by the constant. For instance, differentiate 3x⁵: derivative is 3 × 5x⁴ = 15x⁴.

常数倍法则告诉你，若 y = k u(x)，则 dy/dx = k × du/dx。也就是说，你可以先对变量部分求导，再乘以常数。例如对 3x⁵ 求导：导数为 3 × 5x⁴ = 15x⁴。

5. Finding the Gradient at a Specific Point | 求特定点处的梯度

Once you have the derivative function dy/dx, you can find the gradient of the original curve at any x-coordinate by substituting the x-value into the derivative. This gives the slope of the tangent line at that precise point.

一旦你求出导函数 dy/dx，就可以把任意 x 坐标代入导数，从而得到原曲线在该点的梯度。这就给出了该点切线的斜率。

For example, the curve y = x³ – 3x has derivative dy/dx = 3x² – 3. At x = 2, gradient = 3(2)² – 3 = 12 – 3 = 9. At x = -1, gradient = 3(-1)² – 3 = 0. A zero gradient means the tangent is horizontal, which often indicates a stationary point.

例如，曲线 y = x³ – 3x 的导数是 dy/dx = 3x² – 3。在 x = 2 处，梯度 = 3(2)² – 3 = 12 – 3 = 9。在 x = -1 处，梯度 = 3(-1)² – 3 = 0。梯度为零意味着切线是水平的，这通常表明该点是一个驻点。

6. Equation of a Tangent Line | 切线方程

The tangent to a curve at a given point is a straight line that just touches the curve without crossing it at that point. Its gradient equals the derivative evaluated at the x-coordinate. You can find its equation using the point–slope form: y – y₁ = m(x – x₁), where m = dy/dx at that point.

曲线在给定点处的切线是一条刚好触及曲线且在该点不与之相交的直线。它的斜率等于在该点求得的导数值。你可以使用点斜式求其方程：y – y₁ = m(x – x₁)，其中 m 就是该点的 dy/dx 值。

Example: For y = x² + 4x + 1, find the tangent at x = 1. First, dy/dx = 2x + 4. At x = 1, gradient m = 2(1) + 4 = 6. The point is (1, 1² + 4(1) + 1) = (1, 6). The tangent equation is y – 6 = 6(x – 1), which simplifies to y = 6x.

例题：对于 y = x² + 4x + 1，求 x = 1 处的切线。首先，dy/dx = 2x + 4。x = 1 时，梯度 m = 2(1) + 4 = 6。该点坐标是 (1, 1² + 4(1) + 1) = (1, 6)。切线方程为 y – 6 = 6(x – 1)，化简得 y = 6x。

7. Equation of a Normal Line | 法线方程

The normal line to a curve at a point is perpendicular to the tangent at that same point. If the tangent has gradient m, the normal has gradient -1/m. Be careful: if m = 0, the normal is a vertical line (equation x = constant).

曲线在某点的法线是垂直于该点切线的一条直线。如果切线的斜率为 m，法线的斜率就是 -1/m。注意：若 m = 0，法线是垂直线（方程为 x = 常数）。

Using the previous example, at x = 1 the tangent gradient was 6, so the normal gradient mₙ = -1/6. The point remains (1, 6). The normal equation is y – 6 = -1/6 (x – 1), or y = -1/6 x + 37/6.

沿用前例，x = 1 处切线斜率为 6，因此法线斜率 mₙ = -1/6。点仍然是 (1, 6)。法线方程为 y – 6 = -1/6 (x – 1)，即 y = -1/6 x + 37/6。

8. The Second Derivative | 二阶导数

The second derivative, written as d²y/dx² or f”(x), is obtained by differentiating the first derivative again. It tells us about the rate of change of the gradient itself – in other words, the curvature or concavity of the original function.

二阶导数，写作 d²y/dx² 或 f”(x)，是通过对一阶导数再次求导得到的。它告诉我们梯度本身的变化率，也就是原函数的弯曲程度或凹凸性。

For y = x³ – 3x, the first derivative is dy/dx = 3x² – 3. The second derivative is d²y/dx² = 6x. You can evaluate the second derivative at a specific x to determine whether a stationary point is a maximum or a minimum (the second derivative test).

对于 y = x³ – 3x，一阶导数是 dy/dx = 3x² – 3。二阶导数是 d²y/dx² = 6x。你可以计算特定 x 处的二阶导数值，来判断一个驻点是极大值点还是极小值点（即二阶导数检验法）。

9. Stationary Points and Their Nature | 驻点及其性质

A stationary point occurs where the first derivative is zero, dy/dx = 0. At such a point the tangent is horizontal. There are three types of stationary points: local maximum (peak), local minimum (valley), and point of inflection (where the curve bends but does not change direction).

驻点出现在一阶导数为零，即 dy/dx = 0 的地方。在这样的点处切线是水平的。驻点有三种类型：局部极大值（峰）、局部极小值（谷）以及拐点（曲线在此转向但不改变升降趋势）。

To find stationary points, solve dy/dx = 0 for x. Then substitute these x-values back into the original function to find the corresponding y-coordinates. The nature (max/min/inflection) can be determined using either the first derivative test or the second derivative test.

要找驻点，先解方程 dy/dx = 0 求出 x 值。再将这些 x 值代回原函数，求出对应的 y 坐标。驻点的性质（极大/极小/拐点）可以用一阶导数检验法或二阶导数检验法来判断。

10. The Second Derivative Test for Maxima and Minima | 用二阶导数检验极大值和极小值

The second derivative test is usually the quickest method for classifying stationary points. After finding a stationary point at x = a, evaluate d²y/dx² at x = a. If f”(a) < 0, the point is a maximum; if f''(a) > 0, it is a minimum; if f”(a) = 0, the test is inconclusive.

二阶导数检验法通常是区分驻点类型最快的方法。找到 x = a 处的驻点后，计算 x = a 时的 d²y/dx² 值。若 f”(a) < 0，则是极大值点；若 f''(a) > 0，则是极小值点；若 f”(a) = 0，则该方法无法判断。

Worked example: y = x³ – 3x² – 9x + 5. dy/dx = 3x² – 6x – 9 = 3(x² – 2x – 3) = 3(x – 3)(x + 1). Stationary points at x = 3 and x = -1. Second derivative: d²y/dx² = 6x – 6. At x = 3, d²y/dx² = 12 > 0, so minimum; at x = -1, d²y/dx² = -12 < 0, so maximum. Calculate y: min at (3, -22), max at (-1, 10).

例题：y = x³ – 3x² – 9x + 5。dy/dx = 3x² – 6x – 9 = 3(x² – 2x – 3) = 3(x – 3)(x + 1)。驻点在 x = 3 和 x = -1。二阶导数：d²y/dx² = 6x – 6。在 x = 3 处，d²y/dx² = 12 > 0，因此是极小值点；在 x = -1 处，d²y/dx² = -12 < 0，因此是极大值点。计算 y 值：极小值点 (3, -22)，极大值点 (-1, 10)。

11. Practical Applications – Optimisation | 实际应用——优化问题

Differentiation is frequently used to solve optimisation problems: finding the maximum or minimum value of a quantity such as area, volume, or cost. The approach is to express the quantity as a function of one variable, differentiate, set the derivative to zero, and confirm the nature of the stationary point.

微分常被用来解决优化问题：求诸如面积、体积或成本等量的最大值或最小值。方法是先用一个变量表示该量，求导，令导数等于零，然后确认驻点的性质。

For instance, a rectangular field has 200 m of fencing. One side is bounded by a river and needs no fence, so only three sides require fencing. Maximise the area. Let x be the width (two sides) and y be the length parallel to the river. Perimeter: 2x + y = 200, so y = 200 – 2x. Area A = x y = x(200 – 2x) = 200x – 2x². Differentiate: dA/dx = 200 – 4x = 0 ⇒ x = 50. Second derivative d²A/dx² = -4 < 0, so it's a maximum. Maximum area: A = 50 × (200 - 100) = 5000 m².

例如，一块矩形场地有 200 米围栏。一边临河无需围栏，因此只需围三边。求最大面积。设宽度为 x（两条边），平行于河的边长为 y。周长：2x + y = 200，所以 y = 200 – 2x。面积 A = x y = x(200 – 2x) = 200x – 2x²。求导：dA/dx = 200 – 4x = 0 ⇒ x = 50。二阶导数 d²A/dx² = -4 < 0，所以是极大值。最大面积为 A = 50 × (200 - 100) = 5000 m²。

12. Common Pitfalls and Revision Tips | 常见错误与复习技巧

Many students forget to multiply by the coefficient when using the power rule, or they misapply the sum/difference rule by trying to differentiate the whole expression as one product. Always differentiate term by term. Also, don’t forget that the derivative of a constant is zero.

许多学生在使用幂法则时会忘记乘以系数，或者错误地将整个表达式当作一个乘积来求导。一定要逐项求导。另外，别忘记常数的导数为零。

Another common error is confusing stationary points with points of inflection. Using the second derivative test carefully will help you avoid this. If the second derivative is zero, you must use the first derivative test (check the sign of dy/dx on either side) to determine the nature.

另一个常见错误是混淆驻点与拐点。仔细使用二阶导数检验法能帮助你避免这个错误。如果二阶导数为零，就必须用一阶导数检验法（检验该点两侧 dy/dx 的正负）来判定性质。

Practice writing tangents and normals in the simplest forms (y = mx + c) unless the question asks otherwise. Always sketch a rough graph when solving optimisation problems, as it will help you notice whether your answer makes sense in context.

练习将切线和法线方程写成最简形式（y = mx + c），除非题目另有要求。解优化问题时，一定要画一个粗略的草图，这能帮助你在具体情境中判断答案是否合理。

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IGCSE WJEC Mathematics: Differentiation – Key Concepts and Revision Notes | IGCSE WJEC 数学：微分 考点精讲