IGCSE WJEC Maths: Differentiation – Key Exam Points | IGCSE WJEC 数学：偏微分考点精讲

📚 IGCSE WJEC Maths: Differentiation – Key Exam Points | IGCSE WJEC 数学：偏微分考点精讲

In IGCSE WJEC Mathematics, the term ‘partial differentiation’ is sometimes mistakenly used to refer to ordinary differentiation. Actual partial differentiation involves functions of several variables and is an A-Level or university topic. This article focuses on the differentiation (derivatives) topics that appear in WJEC IGCSE Maths exams, covering every essential point: gradient of a curve, tangents, stationary points, optimisation and kinematics. We will also briefly clarify what partial differentiation really means at the end.

在IGCSE WJEC数学中，“偏微分”一词有时会被误用来指代普通微分。真正的偏微分研究多元函数，是A-Level或大学课程的内容。本文聚焦WJEC IGCSE数学考试中出现的微分（导数）考点，涵盖所有重点：曲线斜率、切线、驻点、最优化以及运动学，并在最后简要说明偏微分的真正含义。

1. Derivative Definition & Notation | 导数定义与符号

The derivative of a function f(x) measures the rate at which the function’s value changes as x changes. Geometrically, it gives the gradient of the tangent to the curve y = f(x). The formal definition uses the limit of a difference quotient:

导数衡量函数 f(x) 的值随 x 变化的快慢。从几何上看，它给出了曲线 y = f(x) 上切线的斜率。正式定义借助差商的极限：

f'(x) = lim_h→0 [f(x+h) – f(x)] / h

Common notations for the derivative include f'(x), dy/dx, y’ and d/dx f(x). In WJEC IGCSE, you will mainly work with dy/dx when the function is given as y = … and with f'(x) when the function is named f(x).

导数的常用记号有 f'(x)、dy/dx、y’ 以及 d/dx f(x)。在 WJEC IGCSE 中，当你遇到 y = … 形式时主要使用 dy/dx，若是 f(x) 函数则常用 f'(x)。

2. Basic Differentiation Rules | 基本微分法则

To differentiate efficiently, you must memorise these four foundation rules. They allow you to handle all polynomial functions tested at IGCSE level.

为了高效求导，你必须熟记下面四条基本法则。它们能处理IGCSE阶段所有的多项式函数。

Constant rule: d/dx (c) = 0, where c is any constant. The graph of a constant is a horizontal line, so its gradient is zero.

常数法则：d/dx (c) = 0，其中 c 为任意常数。常数的图像是一条水平线，因此斜率为零。

Power rule: d/dx (xⁿ) = n xⁿ⁻¹. Multiply by the power, then reduce the power by 1. This works for any real number n, but IGCSE focuses on integer and simple fractional powers.

幂法则：d/dx (xⁿ) = n xⁿ⁻¹。将指数乘到前面，然后把指数减 1。该法则对任意实数 n 成立，但 IGCSE 重点考查整数和简单分数指数。

Constant multiple rule: d/dx (a f(x)) = a f'(x). You can factor out constants before differentiating.

常数倍法则：d/dx (a f(x)) = a f'(x)。求导前可以先将常数提取出来。

Sum or difference rule: d/dx (f(x) ± g(x)) = f'(x) ± g'(x). Differentiate term by term.

和差法则：d/dx (f(x) ± g(x)) = f'(x) ± g'(x)。逐项求导即可。

3. Differentiating Polynomials | 多项式求导

A typical WJEC question asks you to find the derivative of a polynomial like y = 4x³ – 5x² + 7x – 2. Apply the basic rules term by term:

一道典型的WJEC题目会要求你求多项式 y = 4x³ – 5x² + 7x – 2 的导数。逐项应用基本法则即可：

dy/dx = 12x² – 10x + 7

Always remember to rewrite terms with negative or fractional exponents before differentiating, for example 1/x² becomes x⁻², and √x becomes x^½. Then apply the power rule as usual.

务必记住在求导前将带有负指数或分数指数的项改写，例如将 1/x² 写成 x⁻²，√x 写成 x^½，然后再按常规幂法则处理。

4. Gradient of a Curve & Tangent Lines | 曲线斜率与切线

Once you have dy/dx, you can find the gradient of the curve at any specific point by substituting the x‑coordinate. The gradient of the tangent at that point equals the value of the derivative.

求出 dy/dx 后，代入 x 坐标就能得到曲线在某一点处的斜率。该点处的切线斜率就等于导数值。

To obtain the equation of the tangent, use the point‑gradient form y – y₁ = m(x – x₁), where m = dy/dx evaluated at the point (x₁, y₁). A very common exam task is to find the equation of a tangent, or of the normal line which is perpendicular to the tangent (gradient = –1/m).

求切线方程时，利用点斜式 y – y₁ = m(x – x₁)，其中 m 是在点 (x₁, y₁) 处求出的导数值。一个非常热门的考题是求切线方程，或者求法线方程——法线与切线垂直，其斜率为 –1/m。

5. Increasing and Decreasing Functions | 递增与递减函数

The sign of the derivative tells you how the original function behaves. If f'(x) > 0 for all x in an interval, the function is strictly increasing there. If f'(x) < 0, it is strictly decreasing.

导数的符号揭示了原函数的增减性。如果在某个区间内恒有 f'(x) > 0，那么函数在该区间严格递增；若 f'(x) < 0，则严格递减。

You may be asked to find the interval where a curve is increasing by solving an inequality like f'(x) > 0. Sketching a quick sign diagram of f'(x) often helps avoid mistakes.

你可能会被要求解不等式 f'(x) > 0，从而找到曲线递增的区间。快速画出 f'(x) 的符号图往往有助于避免错误。

6. Stationary Points: Maxima & Minima | 驻点：极大与极小值

A stationary point occurs where the derivative is zero: dy/dx = 0. At such a point the tangent is horizontal, and the function could have a local maximum, a local minimum, or a point of inflection.

驻点出现在导数为零处：dy/dx = 0。此时切线水平，函数可能取得局部极大值、局部极小值，或者出现拐点。

To locate stationary points, set f'(x) = 0 and solve for x. Then substitute back into the original equation to find the corresponding y‑coordinates. Remember to give both coordinates in your answer.

要求驻点，先令 f'(x) = 0 解出 x，再代回原方程求出对应的 y 坐标。作答时记得给出完整的 (x, y) 坐标。

You can determine the nature of a stationary point by examining the sign of f'(x) either side of it, or by using the second derivative (see the next section).

判断驻点性质的方法有两种：观察 f'(x) 在驻点两侧的正负号变化，或者使用二阶导数（见下一节）。

7. Second Derivative & Nature of Stationary Points | 二阶导数与驻点性质

The second derivative, denoted f”(x) or d²y/dx², is obtained by differentiating f'(x). It measures the rate of change of the gradient, and tells you whether a curve is concave up or concave down.

二阶导数记作 f”(x) 或 d²y/dx²，由 f'(x) 再次求导得到。它衡量斜率的变化率，并揭示曲线是凹向上还是凹向下。

To classify a stationary point at x = a:

驻点 x = a 的性质判别方法：

If f”(a) > 0, the gradient is increasing; the point is a local minimum.
若 f”(a) > 0， 斜率正在增大；该点为局部极小值。
If f”(a) < 0, the gradient is decreasing; the point is a local maximum.
若 f”(a) < 0， 斜率正在减小；该点为局部极大值。
If f”(a) = 0, the test is inconclusive; use the first‑derivative sign test.
若 f”(a) = 0， 此法无法判断；须用一阶导数符号变化进行检验。

A point of inflection is where the curve changes from concave up to concave down (or vice versa) and f”(x) = 0 or is undefined, but the gradient does not necessarily have to be zero.

拐点是曲线凹凸性改变的位置，此时 f”(x) = 0 或不存在，但一阶导数不一定为零。

8. Kinematics: Velocity and Acceleration | 运动学：速度与加速度

In WJEC IGCSE, differentiation is often applied to the motion of a particle moving in a straight line. If displacement s (in metres) is given as a function of time t (in seconds), then:

在 WJEC IGCSE 中，微分常用于描述沿直线运动的质点。若位移 s（以米为单位）表示为时间 t（以秒为单位）的函数，则有：

v = ds/dt, a = dv/dt = d²s/dt²

Velocity v is the rate of change of displacement, and acceleration a is the rate of change of velocity. A common problem involves finding when a particle is instantaneously at rest (v = 0) or the maximum displacement.

速度 v 是位移的变化率，加速度 a 是速度的变化率。常见问题包括求质点何时瞬时静止 (v = 0) 或何时达到最大位移。

9. Optimisation Problems | 最优化问题

Optimisation requires you to model a real‑world situation with a function, then use differentiation to find its maximum or minimum value. This is a key skill tested in WJEC IGCSE, often involving areas, volumes, or cost functions.

最优化要求你用一个函数模拟实际问题，再通过微分求出最大值或最小值。这是 WJEC IGCSE 重点考查的技能，经常涉及面积、体积或成本函数。

Steps: 1) Express the quantity to be maximised/minimised as a function of one variable. 2) Differentiate and set f'(x) = 0 to find critical values. 3) Use the second‑derivative test or a sign test to confirm the nature. 4) Interpret your answer in the context of the problem, including units.

步骤：1) 把需要优化的量表示成单变量函数。2) 求导并令 f'(x) = 0 找到临界值。3) 使用二阶导数检验或符号检验确认性质。4) 结合题目背景解释答案，记得带上单位。

A classic example: a farmer wants to fence a rectangular paddock using an existing wall for one side. If he has 100 m of fencing, what dimensions give maximum area? Let the dimension parallel to the wall be x, and the other side be y = (100 – x)/2. Express area A = x y, then differentiate with respect to x.

经典例题：农场主想利用一面已有的墙围一个矩形牧场，他一共有 100 m 篱笆，什么尺寸能让面积最大？设与墙平行的边长为 x，另一边长为 y = (100 – x)/2。将面积 A = x y 写成 x 的函数，再对 x 求导。

10. A Note on Partial Differentiation | 关于偏微分的说明

True partial differentiation, written with the symbol ∂, deals with functions of more than one variable, e.g. f(x, y) = x²y + y³. The partial derivative with respect to x treats y as a constant: ∂f/∂x = 2xy, while ∂f/∂y = x² + 3y². This topic is not covered in IGCSE or GCSE; it appears in A‑Level Further Mathematics and university calculus. The term ‘partial differentiation’ is sometimes used informally to mean ordinary differentiation, but you now know the difference.

真正的偏微分使用符号 ∂，研究的是多元函数，例如 f(x, y) = x²y + y³。对 x 的偏导数将 y 视为常数：∂f/∂x = 2xy，而对 y 的偏导数为 ∂f/∂y = x² + 3y²。这一内容不在 IGCSE 或 GCSE 大纲内；它出现在 A‑Level 进阶数学和大学微积分中。“偏微分”一词有时被非正式地用来指普通微分，但你现在已经了解了两者的区别。

When you see a WJEC IGCSE question about ‘finding the gradient’ or ‘turning points’, just recall the straightforward differentiation rules above. Keep practising polynomial differentiation, tangent equations and optimisation, and you will be well prepared for the exam.

当你看到 WJEC IGCSE 中关于“求斜率”或“求极值点”的题目时，只需回忆上文中的普通微分规则即可。多加练习多项式求导、切线方程与最优化问题，你就能为考试做好充分准备。

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IGCSE WJEC Maths: Differentiation – Key Exam Points | IGCSE WJEC 数学：偏微分 考点精讲