GCSE CIE Maths: Partial Derivatives Essentials | GCSE CIE 数学：偏微分考点精讲

📚 GCSE CIE Maths: Partial Derivatives Essentials | GCSE CIE 数学：偏微分考点精讲

Partial derivatives may seem advanced for GCSE level, but they build directly on the ordinary differentiation skills you already possess. Understanding how functions of two or more variables change in different directions provides a powerful link to real-world problems and higher-level mathematics. This guide will walk you through the core concepts, notation, and techniques step by step, using clear examples that connect back to your existing calculus knowledge.

偏导数对于 GCSE 阶段来说也许显得高深，但它直接建立在你们已经掌握的普通微分技巧之上。理解二元或多元函数在不同方向上的变化，能为现实世界的问题和更高阶的数学搭建一座有力的桥梁。本指南将通过清晰的示例，一步步带你掌握核心概念、记号和计算方法，并与你已经具备的微积分知识紧密衔接。

1. Introduction to Multivariable Functions | 多元函数简介

A function of a single variable, such as f(x) = x², takes one input and gives one output. In many real situations, an outcome depends on more than one factor. A multivariable function, like f(x, y) = x² + xy + y², involves two (or more) independent variables.

单变量函数，例如 f(x) = x²，接受一个输入并给出一个输出。在许多实际情况中，结果依赖于不止一个因素。多元函数，像 f(x, y) = x² + xy + y²，就包含两个（或更多）自变量。

You can think of the value f(x, y) as a surface above the xy‑plane: for each pair (x, y), the height of the surface is given by the function. This geometric picture helps when imagining rates of change in different directions.

你可以把 f(x, y) 的值想象为 xy 平面上方的一个曲面：对于每一组 (x, y)，曲面的高度由该函数给出。这一几何图像有助于想象沿不同方向的变化率。

2. What is a Partial Derivative? | 什么是偏导数？

A partial derivative measures how a multivariable function changes when you vary just one of its input variables, while keeping all the others fixed. It is the natural extension of an ordinary derivative to higher dimensions.

偏导数衡量的是，当你只改变多元函数中某一个输入变量而固定所有其他变量时，函数如何变化。它是普通导数向高维空间的自然延伸。

For a function f(x, y), the partial derivative with respect to x, written ∂f/∂x, tells you the instantaneous rate of change of f as x increases and y is held constant. Geometrically, it gives the slope of the surface in the x‑direction at a point.

对于函数 f(x, y)，关于 x 的偏导数，记作 ∂f/∂x，表示在 y 保持不变的情况下，f 随 x 增加时的瞬时变化率。从几何上看，它给出了曲面上某一点沿 x 方向的斜率。

3. Notation and Symbols | 记法与符号

There are several common ways to write partial derivatives. The most frequently used is the ‘curly d’ notation introduced by Legendre:

有几种常见的偏导数记法。最常用的是勒让德引入的“弯 d”符号：

∂f/∂x or ∂/∂x f(x, y)

∂f/∂x 或 ∂/∂x f(x, y)

You may also see subscript notation: fₓ or fₓ(x, y). Both styles mean exactly the same thing – the partial derivative with respect to x. For second‑order derivatives, we write ∂²f/∂x² or fₓₓ, and for mixed partials ∂²f/∂y∂x or fₓₐ, with the order of differentiation read from right to left in the Leibniz form.

你也可能看到下标记法：fₓ 或 fₓ(x, y)。这两种风格含义完全相同——即关于 x 的偏导数。对于二阶导数，我们写 ∂²f/∂x² 或 fₓₓ，对于混合偏导数则写 ∂²f/∂y∂x 或 fₓₐ，莱布尼茨记法中求导顺序是从右向左读。

4. How to Compute Partial Derivatives | 如何计算偏导数

Computing a partial derivative is remarkably simple: you treat all other independent variables as constants and differentiate with respect to the chosen variable using the same rules you already know (power rule, product rule, chain rule, etc.).

计算偏导数非常简单：你把所有其他自变量看作常数，然后用你已经熟知的规则（幂法则、乘法法则、链式法则等）对所选择的变量进行微分。

For example, given f(x, y) = 3x²y + xy³, to find ∂f/∂x, treat y as a constant. Then ∂f/∂x = 3·2x·y + 1·y³ = 6xy + y³. For ∂f/∂y, treat x as a constant: ∂f/∂y = 3x²·1 + x·3y² = 3x² + 3xy².

例如，给定 f(x, y) = 3x²y + xy³，求 ∂f/∂x 时，把 y 当作常数。于是 ∂f/∂x = 3·2x·y + 1·y³ = 6xy + y³。求 ∂f/∂y 时，把 x 当作常数：∂f/∂y = 3x²·1 + x·3y² = 3x² + 3xy²。

Always remember: when differentiating with respect to one variable, any term that does not contain that variable has a derivative of zero.

始终牢记：在对某一个变量求导时，任何不包含该变量的项，其导数均为零。

5. Worked Examples | 详细例题

Let us practise with a few carefully chosen functions. Work through each step to build confidence.

让我们用几个精心挑选的函数来练习。逐步完成每一步，以建立信心。

Example 1: f(x, y) = x³ + 2x²y – y²

例1：f(x, y) = x³ + 2x²y – y²

∂f/∂x: treat y as constant → 3x² + 4xy + 0 = 3x² + 4xy
∂f/∂y: treat x as constant → 0 + 2x² – 2y = 2x² – 2y

∂f/∂x：把 y 看作常数 → 3x² + 4xy + 0 = 3x² + 4xy
∂f/∂y：把 x 看作常数 → 0 + 2x² – 2y = 2x² – 2y

Example 2: f(x, y) = eˣ sin y (the exponential eˣ and sin y)

例2：f(x, y) = eˣ sin y

∂f/∂x: sin y is constant → eˣ sin y
∂f/∂y: eˣ is constant → eˣ cos y

∂f/∂x：sin y 是常数 → eˣ sin y
∂f/∂y：eˣ 是常数 → eˣ cos y

Example 3: f(x, y) = (x + y)/(x – y), for x ≠ y

例3：f(x, y) = (x + y)/(x – y)，其中 x ≠ y

Use the quotient rule: ∂f/∂x = [1·(x – y) – (x + y)·1] / (x – y)² = (x – y – x – y)/(x – y)² = –2y/(x – y)²
∂f/∂y = [1·(x – y) – (x + y)(–1)] / (x – y)² = (x – y + x + y)/(x – y)² = 2x/(x – y)²

使用商法则：∂f/∂x = [1·(x – y) – (x + y)·1] / (x – y)² = (x – y – x – y)/(x – y)² = –2y/(x – y)²
∂f/∂y = [1·(x – y) – (x + y)(–1)] / (x – y)² = (x – y + x + y)/(x – y)² = 2x/(x – y)²

6. Higher‑Order Partial Derivatives | 高阶偏导数

Just as you can differentiate an ordinary derivative to get a second derivative, you can differentiate a partial derivative again to obtain a higher‑order partial derivative. For f(x, y), we can compute ∂²f/∂x², ∂²f/∂y², and the mixed partials ∂²f/∂x∂y and ∂²f/∂y∂x.

就像你可以对普通导数再求导得到二阶导数一样，你也可以对偏导数再次求导以获得高阶偏导数。对于 f(x, y)，我们可以计算 ∂²f/∂x²、∂²f/∂y²，以及混合偏导数 ∂²f/∂x∂y 和 ∂²f/∂y∂x。

The mixed partials often turn out to be equal if the function is continuous and has continuous second derivatives: ∂²f/∂x∂y = ∂²f/∂y∂x. This is known as Clairaut’s theorem (or the symmetry of second derivatives).

如果函数连续且具有连续的二阶导数，混合偏导数往往相等：∂²f/∂x∂y = ∂²f/∂y∂x。这被称为克莱罗定理（或二阶导数的对称性）。

For f(x, y) = x³y + y², we first find ∂f/∂x = 3x²y, then ∂²f/∂x² = 6xy. For ∂²f/∂x∂y, start with ∂f/∂x = 3x²y, then differentiate with respect to y: 3x². If you start with ∂f/∂y = x³ + 2y, then ∂²f/∂y∂x = 3x², confirming the equality.

对于 f(x, y) = x³y + y²，先求出 ∂f/∂x = 3x²y，再得 ∂²f/∂x² = 6xy。对于 ∂²f/∂x∂y，从 ∂f/∂x = 3x²y 出发，对 y 求导得 3x²。如果从 ∂f/∂y = x³ + 2y 出发，∂²f/∂y∂x = 3x²，验证了相等性。

7. The Chain Rule with Partial Derivatives | 偏导数的链式法则

When variables themselves depend on other quantities, the chain rule becomes essential. If z = f(x, y), and both x and y are functions of a single parameter t, then the total derivative of z with respect to t is given by:

当变量本身又依赖于其他量时，链式法则就变得至关重要。如果 z = f(x, y)，且 x 和 y 都是单个参数 t 的函数，那么 z 对 t 的全导数由下式给出：

dz/dt = ∂z/∂x · dx/dt + ∂z/∂y · dy/dt

This formula adds up the rates of change through each intermediate variable. For example, if z = x² + y², and x = cos t, y = sin t, then ∂z/∂x = 2x, ∂z/∂y = 2y, dx/dt = –sin t, dy/dt = cos t. Substituting gives dz/dt = 2x(–sin t) + 2y(cos t) = –2 cos t sin t + 2 sin t cos t = 0, which is expected because x² + y² = 1 for all t.

这一公式将经过每个中间变量的变化率相加。例如，若 z = x² + y²，x = cos t，y = sin t，则 ∂z/∂x = 2x，∂z/∂y = 2y，dx/dt = –sin t，dy/dt = cos t。代入得到 dz/dt = 2x(–sin t) + 2y(cos t) = –2 cos t sin t + 2 sin t cos t = 0，这正是预期的结果，因为对所有 t 都有 x² + y² = 1。

If x and y themselves depend on two parameters, say u and v, then the chain rule extends naturally. You can use a tree diagram to keep track of the dependencies when applying the rule.

如果 x 和 y 本身又依赖于两个参数，例如 u 和 v，那么链式法则自然随之扩展。应用时，你可以使用树形图来跟踪依赖关系。

8. Applications of Partial Derivatives | 偏导数的应用

Partial derivatives appear in many areas of science and engineering. They are used to find maxima and minima of surfaces, optimise functions subject to constraints (Lagrange multipliers), and in the study of rates of change in thermodynamics, economics, and fluid dynamics.

偏导数出现在许多科学与工程领域中。它们被用来求曲面的极大值和极小值、在约束条件下优化函数（拉格朗日乘数法），以及在热力学、经济学和流体动力学等学科中研究变化率。

In economics, a production function Q(K, L) might represent output as a function of capital K and labour L. The partial derivative ∂Q/∂K is the marginal product of capital – the extra output from a small increase in capital, holding labour constant.

在经济学中，生产函数 Q(K, L) 可以表示产出作为资本 K 和劳动 L 的函数。偏导数 ∂Q/∂K 就是资本的边际产量——在劳动保持不变的情况下，资本少量增加所带来的额外产出。

In physics, the heat equation ∂u/∂t = α ∂²u/∂x² describes how temperature u(x, t) evolves over time. Here, a partial differential equation links changes in temperature with respect to both time and position.

在物理学中，热方程 ∂u/∂t = α ∂²u/∂x² 描述了温度 u(x, t) 随时间如何变化。在这里，偏微分方程将温度随时间和位置的变化联系在一起。

9. Common Mistakes and How to Avoid Them | 常见错误与避免方法

Mistake 1: Forgetting to treat the fixed variable as a constant. When finding ∂f/∂x, every term involving y alone or constants must be differentiated as if y were a number. For instance, in f(x, y) = x sin y + y², ∂f/∂x = sin y (y² becomes 0). Do not let the presence of y trick you into applying the product rule unnecessarily.

错误1： 忘记把固定变量当作常数。求 ∂f/∂x 时，任何只含 y 或常数的项必须被当作数字来求导。例如，在 f(x, y) = x sin y + y² 中，∂f/∂x = sin y（y² 变为 0）。不要让 y 的存在诱使你无谓地使用乘法法则。

Mistake 2: Mixing up the order of mixed partials notation. Remember that ∂²f/∂x∂y means first differentiate with respect to x, then y. In subscript form, fₓₐ means (fₓ)ₐ, i.e. first x, then y – consistent with the denominator reading right‑to‑left.

错误2： 混淆混合偏导数的记法顺序。请记住 ∂²f/∂x∂y 意味着先对 x 求导，再对 y 求导。在下标形式中，fₓₐ 表示 (fₓ)ₐ，即先 x 后 y——与分母从右向左读的顺序一致。

Mistake 3: Over‑generalising rules from single‑variable calculus. The product rule still works, but you must apply it only to the variable you are differentiating with respect to. If you have f(x, y) = g(x)h(y), then ∂f/∂x = g'(x)h(y) with h(y) kept constant – the product rule does not split h(y) into a derivative.

错误3： 过度推广单变量微积分中的规则。乘法法则依然有效，但你只能把它应用到你正在求导的那个变量上。如果 f(x, y) = g(x)h(y)，那么 ∂f/∂x = g'(x)h(y)，h(y) 保持为常数——乘法法则并不会让 h(y) 产生导数。

10. Practice Questions and Summary | 练习题目与总结

Test your understanding with these problems. Try to solve them without looking at the hints first.

用以下题目来检验你的理解。先不要看提示，尝试自己求解。

Find ∂f/∂x and ∂f/∂y for f(x, y) = 4x³ – 5xy² + 2y⁴.

求 f(x, y) = 4x³ – 5xy² + 2y⁴ 的 ∂f/∂x 和 ∂f/∂y。
For g(x, y) = ln(x² + y²), compute ∂g/∂x and ∂g/∂y.

对于 g(x, y) = ln(x² + y²)，计算 ∂g/∂x 和 ∂g/∂y。
If u = xy + yz + zx, find all first‑order partial derivatives ∂u/∂x, ∂u/∂y, ∂u/∂z.

若 u = xy + yz + zx，求所有一阶偏导数 ∂u/∂x、∂u/∂y、∂u/∂z。
For h(x, y) = eˣʸ, verify that ∂²h/∂x∂y = ∂²h/∂y∂x.

对于 h(x, y) = eˣʸ，验证 ∂²h/∂x∂y = ∂²h/∂y∂x。
Given w = x² + y², where x = r cos θ, y = r sin θ, use the chain rule to find ∂w/∂r and ∂w/∂θ.

已知 w = x² + y²，其中 x = r cos θ，y = r sin θ，用链式法则求 ∂w/∂r 和 ∂w/∂θ。

Partial derivatives are a natural extension of ordinary differentiation to functions of several variables. By treating all but one variable as constant, you can reuse all your existing differentiation skills. Whether you are preparing for advanced studies or simply curious about multi‑dimensional calculus, mastering partial derivatives opens the door to a richer understanding of how quantities relate to one another in a changing world.

偏导数是普通微分对多元函数的自然拓展。通过固定除一个变量以外的所有其他变量，你可以重用所有已经掌握的微分技能。无论你是在为高阶学习做准备，还是对多维微积分感到好奇，掌握偏导数都将为你更深刻地理解各个量在变化世界中的相互关系打开一扇大门。

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GCSE CIE Maths: Partial Derivatives Essentials | GCSE CIE 数学：偏微分考点精讲