A-Level AQA Mathematics: Partial Differentiation Key Concepts | A-Level AQA 数学：偏微分考点精讲

📚 A-Level AQA Mathematics: Partial Differentiation Key Concepts | A-Level AQA 数学：偏微分考点精讲

Partial differentiation extends the ideas of ordinary differentiation to functions of several variables. In A-Level AQA Further Mathematics, you will learn how to compute and apply partial derivatives, handle the chain rule for multivariable functions, find and classify stationary points of surfaces, and use the total differential for error estimation. This guide breaks down every key topic, equips you with essential techniques, and highlights common pitfalls so you can approach exam questions with confidence.

偏微分将普通微分的概念推广到多元函数。在 A-Level AQA 进阶数学中，你需要学习如何计算和应用偏导数，掌握多元函数的链式法则，寻找并分类曲面的驻点，以及运用全微分进行误差估计。本指南将逐一讲解每个核心知识点，配备关键的解题技巧，并提示常见错误，帮助你自信应对考试中的各类问题。

1. Introduction to Partial Differentiation | 偏微分简介

When a function depends on two or more independent variables, say z = f(x, y), the rate of change of z with respect to one variable while keeping the others constant is called a partial derivative.

当一个函数依赖于两个或更多自变量时，例如 z = f(x, y)，将其他变量视为常数，对其中一个变量求导，就得到了偏导数。

In physical contexts, partial derivatives describe how a quantity such as temperature varies with position along one axis, or how the volume of a gas changes with pressure when temperature is fixed. Mastery of partial differentiation is essential for optimisation problems, error analysis, and further study in calculus.

在物理情境中，偏导数可以描述温度如何沿某一个坐标轴变化，或者当温度固定时气体体积如何随压强变化。掌握偏微分对于解决最优化问题、进行误差分析以及学习更高阶的微积分都至关重要。

2. Partial Derivatives: Notation and First Principles | 偏导数：符号与第一原理

For z = f(x, y), the first partial derivatives are written as ∂f/∂x and ∂f/∂y, or simply fₓ and fᵧ. The curly symbol ∂ emphasises that we are dealing with a multivariable function.

对于 z = f(x, y)，一阶偏导数记作 ∂f/∂x 和 ∂f/∂y，或简单地写作 fₓ 与 fᵧ。弯曲符号 ∂ 强调我们正在处理多元函数。

To compute fₓ, treat y as a constant and differentiate with respect to x using standard rules. For example, if f(x, y) = x² sin y + 3xy³, then fₓ = 2x sin y + 3y³, and fᵧ = x² cos y + 9xy².

计算 fₓ 时，将 y 视为常数，用常规求导法则对 x 求导。例如，若 f(x, y) = x² sin y + 3xy³，则 fₓ = 2x sin y + 3y³，fᵧ = x² cos y + 9xy²。

The definition via a limit mirrors ordinary differentiation:

∂f/∂x = limₕ→₀ [f(x+h, y) – f(x, y)] / h

with y held constant. Although you will not often be asked to work directly from this definition, it underpins the geometric meaning — the slope of the tangent line to the surface in the x-direction.

通过极限定义与普通微分类似：

∂f/∂x = limₕ→₀ [f(x+h, y) – f(x, y)] / h

，其中 y 保持不变。虽然你很少需要直接使用这个定义，但它揭示了偏导数的几何意义 —— 曲面沿 x 方向切线斜率。

3. Higher-Order Partial Derivatives | 高阶偏导数

Just as with ordinary derivatives, we can differentiate partial derivatives to obtain second-order partial derivatives. For z = f(x, y), the four possibilities are: fₓₓ = ∂²f/∂x², fᵧᵧ = ∂²f/∂y², fₓᵧ = ∂²f/∂y∂x, and fᵧₓ = ∂²f/∂x∂y.

与普通导数一样，我们可以对偏导数再次求导，得到二阶偏导数。对于 z = f(x, y)，共有四种可能：fₓₓ = ∂²f/∂x², fᵧᵧ = ∂²f/∂y², fₓᵧ = ∂²f/∂y∂x, 以及 fᵧₓ = ∂²f/∂x∂y。

Under mild continuity conditions, the mixed partial derivatives are equal: ∂²f/∂y∂x = ∂²f/∂x∂y. This is Clairaut’s theorem. In AQA exams you can safely assume equality for all functions you encounter unless instructed otherwise.

在适当的连续性条件下，混合偏导数相等：∂²f/∂y∂x = ∂²f/∂x∂y。这就是克莱罗定理。在 AQA 考试中，除非特别说明，你都可以安全地假定所有遇到的函数其混合偏导数相等。

Higher-order derivatives are used to classify stationary points and to solve partial differential equations. Notation such as fₓₓᵧ and fₓᵧᵧ continues the pattern naturally.

高阶偏导数用于分类驻点以及求解偏微分方程。像 fₓₓᵧ 和 fₓᵧᵧ 这样的符号自然地沿用了相同的表示方法。

4. Chain Rule for Partial Derivatives | 偏导数的链式法则

When the variables themselves depend on other parameters, the chain rule for a function f(x, y) where x = x(t) and y = y(t) is:

df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

This is the total derivative with respect to t. It sums the contributions from each variable path.

当变量本身依赖于其他参数时，函数 f(x, y)，其中 x = x(t)、y = y(t) 的链式法则为：

df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

这是关于 t 的全导数，它汇总了来自每个变量路径的贡献。

If instead x and y are functions of two new variables u and v, the chain rule generalises: ∂f/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u), and similarly for ∂f/∂v. This pattern is essential for coordinate transformations, such as from Cartesian to polar coordinates.

若 x 和 y 本身又是两个新变量 u 和 v 的函数，则链式法则推广为：∂f/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u)，∂f/∂v 类似。这一形式在坐标变换（如从直角坐标变为极坐标）中至关重要。

Typical AQA exam questions will give specific forms like x = u² – v, y = uv and ask for ∂f/∂u in terms of the partial derivatives of f with respect to x and y. The key is to identify each branch of dependence and apply the rule systematically.

典型的 AQA 考题会给出具体形式如 x = u² – v, y = uv，并要求用 f 关于 x 和 y 的偏导数表示 ∂f/∂u。关键在于识别每条依赖路径，并系统性地应用该法则。

5. Implicit Partial Differentiation | 隐函数偏微分

For an equation connecting three variables, such as F(x, y, z) = 0, we can find partial derivatives of one variable with respect to the others, treating the remaining variable as constant, by differentiating implicitly.

对于一个关联三个变量的方程，如 F(x, y, z) = 0，我们可以用隐函数微分法求其中某个变量对其余变量的偏导数，并在求导过程中将第三个变量视为常数。

For example, to find ∂z/∂x, differentiate the whole equation with respect to x, treating y as constant, and remembering that z depends on x, so ∂/∂x (z) = ∂z/∂x. Rearrange to solve for ∂z/∂x. The formal formula is:

∂z/∂x = – (∂F/∂x) / (∂F/∂z)

provided ∂F/∂z ≠ 0.

例如，为求 ∂z/∂x，将整个方程对 x 求导，把 y 视为常数，并记住 z 依赖于 x，因此 ∂/∂x (z) = ∂z/∂x。移项解出 ∂z/∂x。其正式公式为：

∂z/∂x = – (∂F/∂x) / (∂F/∂z)

前提是 ∂F/∂z ≠ 0。

This technique is widely applied in thermodynamics and geometry, where relationships between pressure, volume, and temperature, or the defining equation of a surface, are given implicitly.

这一技巧广泛应用于热力学和几何中，常见于压强、体积和温度之间，或定义曲面的隐式方程给出的关系。

6. Total Differential and Small Changes | 全微分与微小变化

If z = f(x, y) and x and y change by small increments Δx and Δy, the corresponding change in z is approximately:

Δz ≈ (∂f/∂x) Δx + (∂f/∂y) Δy

The expression dz = (∂f/∂x) dx + (∂f/∂y) dy is called the total differential. It is the linear approximation of the change in f.

若 z = f(x, y)，且 x 和 y 发生微小增量 Δx 和 Δy，则 z 相应的变化可近似为：

Δz ≈ (∂f/∂x) Δx + (∂f/∂y) Δy

表达式 dz = (∂f/∂x) dx + (∂f/∂y) dy 称为全微分。它是 f 变化的线性近似。

This formula is used to estimate errors. If Δx and Δy represent maximum possible errors in measurements, the maximum absolute error in z is often estimated as |∂f/∂x| |Δx| + |∂f/∂y| |Δy|. Many AQA questions ask you to calculate the approximate change in volume of a cylinder when radius and height are measured with small uncertainties.

该公式用于误差估计。如果 Δx 和 Δy 代表测量中的最大可能误差，那么 z 的最大绝对误差通常可估算为 |∂f/∂x| |Δx| + |∂f/∂y| |Δy|。很多 AQA 试题要求你计算当圆柱的半径和高带有微小不确定度时，其体积的近似变化。

Remember that the approximation improves as Δx and Δy become smaller, and it relies on the differentiability of f.

请记住，当 Δx 和 Δy 越小时，该近似越精确，且它的成立依赖于 f 的可微性。

7. Stationary Points of Functions of Two Variables | 二元函数的驻点

A function f(x, y) has a stationary point where both first partial derivatives vanish simultaneously: fₓ = 0 and fᵧ = 0. These points can correspond to local maxima, local minima, or saddle points.

函数 f(x, y) 的驻点是一阶偏导数同时为零的点：fₓ = 0 且 fᵧ = 0。这些点可能对应于局部极大值、局部极小值或鞍点。

To find stationary points, solve the system of equations ∂f/∂x = 0 and ∂f/∂y = 0. This often involves simultaneous algebraic manipulation. For example, if f(x, y) = x³ + 3xy + y³, then setting ∂f/∂x = 3x² + 3y = 0 and ∂f/∂y = 3x + 3y² = 0 yields x² = –y and y² = –x, leading to solutions (0,0) and (–1, –1).

要找到驻点，求解方程组 ∂f/∂x = 0 和 ∂f/∂y = 0。这通常会涉及联立代数运算。例如，若 f(x, y) = x³ + 3xy + y³，则令 ∂f/∂x = 3x² + 3y = 0 和 ∂f/∂y = 3x + 3y² = 0，可得 x² = –y 和 y² = –x，进而解出 (0,0) 和 (–1, –1)。

Be methodical: always check both equations are satisfied and present coordinates clearly. A common mistake is to misapply the product rule or to forget that y is treated as a constant when differentiating with respect to x.

请有条理地进行计算：始终检验两个方程是否都成立，并清晰地写出坐标。常见的错误是错误使用乘积法则，或在关于 x 求导时忘记将 y 视为常数。

8. Classification of Stationary Points | 驻点分类

Once stationary points are found, we classify them using second partial derivatives. Define the discriminant:

D = fₓₓ fᵧᵧ – (fₓᵧ)²

Evaluate D at the stationary point. The rules are:

If D > 0 and fₓₓ > 0, the point is a local minimum.
If D > 0 and fₓₓ < 0, the point is a local maximum.
If D < 0, the point is a saddle point.
If D = 0, the test is inconclusive (further investigation needed).

找到驻点后，我们用二阶偏导数对其进行分类。定义判别式：

D = fₓₓ fᵧᵧ – (fₓᵧ)²

在驻点处计算 D。判别规则为：

若 D > 0 且 fₓₓ > 0，该点为局部极小值点。
若 D > 0 且 fₓₓ < 0，该点为局部极大值点。
若 D < 0，该点为鞍点。
若 D = 0，该判别法失效（需进一步分析）。

For the example f(x, y) = x³ + 3xy + y³, we compute fₓₓ = 6x, fᵧᵧ = 6y, fₓᵧ = 3. At (0,0): fₓₓ = 0, fᵧᵧ = 0, D = 0 ⋅ 0 – 3² = –9 < 0, so (0,0) is a saddle point. At (–1, –1): fₓₓ = –6, fᵧᵧ = –6, D = (–6)(–6) – 9 = 27 > 0, and fₓₓ < 0, hence a local maximum.

以前例 f(x, y) = x³ + 3xy + y³ 为例，计算 fₓₓ = 6x, fᵧᵧ = 6y, fₓᵧ = 3。在 (0,0) 处：fₓₓ = 0, fᵧᵧ = 0, D = 0 ⋅ 0 – 3² = –9 < 0，故 (0,0) 为鞍点。在 (–1, –1) 处：fₓₓ = –6, fᵧᵧ = –6, D = (–6)(–6) – 9 = 27 > 0，且 fₓₓ < 0，因此是局部极大值点。

In exams, always state D, the sign of fₓₓ, and a clear conclusion. Never classify a point based only on the sign of fₓₓ or fᵧᵧ alone — the discriminant must be considered first. Mixed derivatives often trip students up, so double-check your differentiation.

在考试中，必须给出 D、fₓₓ 的符号以及清晰的结论。绝不能仅凭 fₓₓ 或 fᵧᵧ 的正负进行分类 —— 必须首先考察判别式。混合导数常让学生犯错，请仔细检查你的求导过程。

9. Applications: Error Analysis | 应用：误差分析

Partial differentiation provides the framework for error propagation. If a quantity Q is calculated from measured values x and y with uncertainties ±Δx and ±Δy, the maximum absolute error in Q is approximated by |∂Q/∂x| Δx + |∂Q/∂y| Δy.

偏微分为误差传递提供了理论框架。若某个量 Q 由测量值 x 和 y 计算得出，且测量值各自带有不确定度 ±Δx 和 ±Δy，则 Q 的最大绝对误差可近似为 |∂Q/∂x| Δx + |∂Q/∂y| Δy。

For a more statistical treatment, the root-sum-square formula √[(∂Q/∂x Δx)² + (∂Q/∂y Δy)²] is used, but AQA questions typically focus on the linear (max error) estimate. Always state whether you are finding the maximum possible error or a percentage error, and show the substitution clearly.

在更偏统计的处理中会使用方和根公式 √[(∂Q/∂x Δx)² + (∂Q/∂y Δy)²]，但 AQA 试题通常关注线性（最大误差）估计。记得清楚说明你是在求最大可能误差还是百分比误差，并清晰地展示代入过程。

A worked example: The volume V of a cone is V = ⅓ πr²h. Given r = 5 cm ± 0.1 cm and h = 10 cm ± 0.2 cm, find the approximate maximum error in V. ∂V/∂r = ⅔ πrh, ∂V/∂h = ⅓ πr². Substituting nominal values gives ΔV ≈ (⅔ π × 5 × 10) × 0.1 + (⅓ π × 25) × 0.2 ≈ (100π/3)×0.1 + (25π/3)×0.2 = (10π/3) + (5π/3) = 5π ≈ 15.71 cm³. This illustrates the direct use of partial derivatives in practical problems.

例题：圆锥体积 V = ⅓ πr²h。已知 r = 5 cm ± 0.1 cm，h = 10 cm ± 0.2 cm，求 V 的近似最大误差。∂V/∂r = ⅔ πrh，∂V/∂h = ⅓ πr²。代入标称值：ΔV ≈ (⅔ π × 5 × 10) × 0.1 + (⅓ π × 25) × 0.2 ≈ (100π/3)×0.1 + (25π/3)×0.2 = (10π/3) + (5π/3) = 5π ≈ 15.71 cm³。该例展示了偏导数在实际问题中的直接应用。

10. Exam Tips and Common Mistakes | 考试技巧与常见错误

✔ Do: Write partial derivative symbols clearly; keep constant variables in mind at every step; simplify expressions before evaluating D; double-check second derivatives, especially mixed ones; always conclude with words like “local minimum” or “saddle point” in classification questions.

✔ 务必：清楚地书写偏导数符号；每一步都牢记哪些变量视为常数；在计算判别式 D 之前先化简表达式；仔细检查二阶导数，尤其是混合导数；在分类题中始终用文字如“局部极小值”或“鞍点”给出结论。

✘ Don’t: Use ordinary derivative notation d/dx when dealing with multivariable functions; forget the product rule when terms contain both x and y; omit the negative sign in implicit partial differentiation; confuse Δz (actual change) with dz (total differential) — the former is exact, the latter an approximation; classify stationary points solely on fₓₓ sign without checking D.

✘ 切勿：在处理多元函数时使用普通导数符号 d/dx；当项中同时含有 x 和 y 时忘记使用乘积法则；在隐函数偏微分中遗漏负号；混淆 Δz（实际变化量）与 dz（全微分）—— 前者是精确值，后者是近似值；仅凭 fₓₓ 符号而忽略 D 来分类驻点。

In the run-up to the exam, practise identifying dependence chains in chain-rule questions. Draw a simple tree diagram: f at the top, then branches to x and y, then from x and y to the final variable. This visual stop errors. Also, when error intervals are given as percentages, convert to absolute increments first.

考前练习时，在链式法则题目中可通过绘制简单的树状图来识别依赖关系：f 在最上方，向下分支到 x 和 y，再从 x 和 y 连接到最终变量。这样的可视化方法可以避免错误。此外，当误差以百分比给出时，应先转换为绝对增量。

Finally, manage your time: partial differentiation questions often require multiple parts, so allocate your minutes according to marks. Show all working; even if the final classification is incorrect, marks are awarded for correct partial derivatives and correct evaluation of D.

最后，合理分配时间：偏微分题目通常包含多个小问，要根据分值分配答题时间。展现所有计算过程；即使最终分类有误，正确的偏导数计算和 D 的求值也能获得相应分数。

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A-Level AQA Mathematics: Partial Differentiation Key Concepts | A-Level AQA 数学：偏微分 考点精讲