Partial Differentiation Key Points | 偏微分考点精讲

📚 Partial Differentiation Key Points | 偏微分考点精讲

Partial differentiation is a fundamental tool in multivariable calculus, enabling us to explore how functions of several variables change with respect to one variable while holding the others constant. This topic appears in advanced high‑school curricula such as IB Mathematics Analysis & Approaches HL and some AQA Further Mathematics modules, forming the basis for optimisation, tangent planes, and error analysis. This article provides a thorough revision guide covering definitions, notation, computation rules, chain rules, higher‑order derivatives, optimisation of two‑variable functions, and common pitfalls.

偏微分是多元微积分的核心工具，用于探究多变量函数在固定其他变量时对某一个变量的变化率。这一主题出现在IB数学分析与方法HL以及部分AQA进阶数学模块中，是优化问题、切平面和误差分析的基础。本文将从定义、符号、计算法则、链式法则、高阶偏导、二元函数极值分类等多个方面进行全面梳理，帮助考生精准把握考点。

1. Definition of a Partial Derivative | 偏导数的定义

Given a function of two variables, f(x,y), the partial derivative with respect to x at a point (a,b) is defined as ∂f/∂x = lim_{h→0} [f(a+h, b) − f(a,b)] / h, provided this limit exists. Geometrically, it represents the slope of the tangent line to the curve obtained by intersecting the surface z = f(x,y) with the plane y = b. Similarly, ∂f/∂y is the rate of change in the y‑direction with x fixed.

对于二元函数 f(x,y)，关于 x 的偏导数定义为 ∂f/∂x = limₕ→₀ [f(a+h, b) − f(a,b)] / h，前提是该极限存在。几何意义上，它表示曲面 z = f(x,y) 与平面 y = b 相交所得曲线在点 (a,b) 处的切线斜率。类似地，∂f/∂y 表示固定 x 时沿 y 方向的变化率。

When we write fₓ or ∂f/∂x, we treat y as a constant and differentiate as if f were a single‑variable function of x only. This ‘partial’ viewpoint extends naturally to functions of three or more variables, e.g., ∂f/∂z for f(x,y,z).

当我们记作 fₓ 或 ∂f/∂x 时，我们将 y 视为常数，然后像对单变量函数求导那样对 x 求导。这种“局部”的观点自然可以推广到三元或更多元的函数，如 f(x,y,z) 的 ∂f/∂z。

2. Notation for Partial Derivatives | 偏导数的符号表示

The most common notation is the curly ‘∂’ introduced by Legendre. For a function z = f(x,y), we write ∂z/∂x or ∂f/∂x, and sometimes fₓ(x,y). Second‑order partial derivatives are denoted ∂²z/∂x², ∂²z/∂y², or ∂²z/∂x∂y for the mixed derivative. Alternative notations include fₓₓ, fᵧᵧ, and fₓᵧ.

最常见的符号是拉格朗日引入的花体 “∂”。对函数 z = f(x,y)，我们记作 ∂z/∂x 或 ∂f/∂x，有时也用 fₓ(x,y)。二阶偏导数记作 ∂²z/∂x²、∂²z/∂y²，以及混合偏导 ∂²z/∂x∂y。替代记法包括 fₓₓ、fᵧᵧ 和 fₓᵧ。

In physics and engineering, subscript notation is prevalent: uₓ = ∂u/∂x. Always pay attention to the order of differentiation in mixed derivatives: ∂²f/∂x∂y means differentiate first with respect to y, then x, whereas fₓᵧ means (fₓ)ᵧ, i.e., differentiate with respect to y after x.

在物理和工程中，下标记法很普遍：uₓ = ∂u/∂x。务必注意混合偏导数的求导顺序：∂²f/∂x∂y 表示先对 y 求导再对 x 求导，而 fₓᵧ 表示先对 x 求导再对 y 求导，即 (fₓ)ᵧ。

3. Basic Rules of Partial Differentiation | 偏微分的基本法则

To compute ∂f/∂x, treat all variables other than x as constants and apply the standard differentiation rules (power rule, product rule, chain rule, etc.). For example, if f(x,y) = x³y + sin(xy), then ∂f/∂x = 3x²y + y cos(xy), and ∂f/∂y = x³ + x cos(xy).

计算 ∂f/∂x 时，将 x 以外的所有变量视作常数，运用常规的求导法则（幂法则、乘法法则、链式法则等）。例如，f(x,y) = x³y + sin(xy)，则 ∂f/∂x = 3x²y + y cos(xy)，∂f/∂y = x³ + x cos(xy)。

Always check for composite functions: when differentiating sin(xy) with respect to x, the derivative of the inner function xy is y (since y is constant). The same logic extends to exponentials and logarithms: ∂/∂x (e^(xy)) = y e^(xy).

遇到复合函数时需要格外小心：对 sin(xy) 关于 x 求导时，内部函数 xy 的导数为 y（因为 y 视为常数）。同样的逻辑适用于指数和对数函数：∂/∂x (e^(xy)) = y e^(xy)。

4. Second and Higher-Order Partial Derivatives | 二阶与高阶偏导数

After obtaining first‑order partial derivatives fₓ and fᵧ, we can differentiate again to obtain second‑order derivatives. There are four possibilities: ∂²f/∂x² = (fₓ)ₓ, ∂²f/∂y² = (fᵧ)ᵧ, and the two mixed derivatives ∂²f/∂x∂y and ∂²f/∂y∂x.

得到一阶偏导 fₓ 和 fᵧ 后，可以继续求导得到二阶偏导数。共有四种可能：∂²f/∂x² = (fₓ)ₓ，∂²f/∂y² = (fᵧ)ᵧ，以及两个混合偏导数 ∂²f/∂x∂y 和 ∂²f/∂y∂x。

Third‑order and higher derivatives follow the same pattern, e.g., fₓₓᵧ = ∂³f/∂y∂x². In practice, you will mostly deal with first and second derivatives when locating and classifying stationary points.

三阶及更高阶的偏导遵循相同的模式，例如 fₓₓᵧ = ∂³f/∂y∂x²。实际应用中，最常见的是在求驻点及其分类时用到的一阶与二阶偏导数。

5. Equality of Mixed Partial Derivatives | 混合偏导数的对称性

Clairaut’s theorem (or Schwarz’s theorem) states that if the second‑order mixed partial derivatives of f are continuous, then ∂²f/∂x∂y = ∂²f/∂y∂x. This symmetry greatly simplifies calculations because you can choose the easier order of differentiation.

克莱罗定理（或施瓦茨定理）指出，如果 f 的二阶混合偏导数连续，则 ∂²f/∂x∂y = ∂²f/∂y∂x。这一对称性大大简化了计算，因为你可以选择更简便的求导顺序。

For example, if f(x,y) = x²eʸ + y cos x, differentiate with respect to x then y: fₓ = 2x eʸ – y sin x, then fₓᵧ = 2x eʸ – sin x. Doing y first, fᵧ = x²eʸ + cos x, then fᵧₓ = 2x eʸ – sin x. They are equal, as expected.

例如，f(x,y) = x²eʸ + y cos x，先对 x 再对 y 求导：fₓ = 2x eʸ – y sin x，然后 fₓᵧ = 2x eʸ – sin x。先对 y 再对 x：fᵧ = x²eʸ + cos x，然后 fᵧₓ = 2x eʸ – sin x。结果相等，符合定理。

6. Total Differential | 全微分

The total differential of z = f(x,y) is dz = (∂f/∂x) dx + (∂f/∂y) dy. It approximates the change in f when x and y change by small increments dx and dy. This is used in linear approximation and error propagation: Δz ≈ fₓ Δx + fᵧ Δy.

函数 z = f(x,y) 的全微分为 dz = (∂f/∂x) dx + (∂f/∂y) dy。它近似表示当 x 和 y 分别发生微小变化 dx 和 dy 时 f 的改变量。全微分常用于线性近似和误差传播：Δz ≈ fₓ Δx + fᵧ Δy。

For example, the volume of a cylinder V = πr²h. If the radius and height are measured with small uncertainties Δr and Δh, the uncertainty in volume is roughly ΔV ≈ 2πrh Δr + πr² Δh. The total differential provides this linearised estimate.

例如，圆柱体积 V = πr²h。若半径和高度存在微小测量误差 Δr 和 Δh，则体积的误差近似为 ΔV ≈ 2πrh Δr + πr² Δh。全微分给出了这一线性化估计。

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

When variables themselves depend on another parameter, say x = x(t) and y = y(t), the rate of change of f(x(t), y(t)) with respect to t is given by df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). This is the chain rule for paths.

当变量本身依赖于另一个参数时，例如 x = x(t)、y = y(t)，f(x(t), y(t)) 对 t 的变化率为 df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)。这是针对路径的链式法则。

A more general form applies when x = x(u,v) and y = y(u,v). Then ∂f/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u), and similarly for v. Drawing a dependency tree helps organise the terms and avoid missing contributions.

更一般的形式适用于 x = x(u,v) 且 y = y(u,v) 的情形。此时 ∂f/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u)，类似地可写出 ∂f/∂v。绘制依赖树有助于整理各项，避免遗漏。

8. Implicit Partial Differentiation | 隐函数偏微分

An equation F(x,y,z) = 0 implicitly defines z as a function of x and y. Assuming Fₓ, Fᵧ, F_z are continuous and F_z ≠ 0, we can find ∂z/∂x = −Fₓ/F_z and ∂z/∂y = −Fᵧ/F_z. These formulas are derived by differentiating the identity F(x,y,z(x,y)) = 0 with respect to x or y.

方程 F(x,y,z) = 0 隐式地定义了 z 是 x 和 y 的函数。假设 Fₓ、Fᵧ、F_z 连续且 F_z ≠ 0，可得 ∂z/∂x = −Fₓ/F_z，∂z/∂y = −Fᵧ/F_z。这些公式通过对恒等式 F(x,y,z(x,y)) = 0 关于 x 或 y 求导得到。

Example: x² + y² + z² = 1. Let F = x² + y² + z² − 1. Then ∂z/∂x = −2x/(2z) = −x/z, and ∂z/∂y = −y/z. This technique avoids solving explicitly for z, which is often messy or impossible.

例如：x² + y² + z² = 1。令 F = x² + y² + z² − 1，则 ∂z/∂x = −2x/(2z) = −x/z，∂z/∂y = −y/z。此技巧避免了显式解出 z，而显式解往往非常复杂甚至不可能。

9. Stationary Points and Classification | 驻点与极值分类

For a function f(x,y), stationary points occur where both first‑order partial derivatives vanish: fₓ = 0 and fᵧ = 0. To classify them, compute the second‑order derivatives and form the discriminant D = fₓₓ fᵧᵧ − (fₓᵧ)².

对于函数 f(x,y)，驻点出现在所有一阶偏导数同时为零处：fₓ = 0 且 fᵧ = 0。为了分类，需要计算二阶导数并构造判别式 D = fₓₓ fᵧᵧ − (fₓᵧ)²。

At a stationary point (a,b):

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 (neither max nor min).
If D = 0, the test is inconclusive; further investigation is needed.

在驻点 (a,b) 处：

若 D > 0 且 fₓₓ > 0，该点为局部极小值。
若 D > 0 且 fₓₓ < 0，该点为局部极大值。
若 D < 0，该点为鞍点（既非极大也非极小）。
若 D = 0，判别法无效，需要进一步分析。

Always remember that D is evaluated at the stationary point. A classic example: f(x,y) = x² − y² gives fₓ = 2x, fᵧ = −2y, stationary at (0,0). Then fₓₓ = 2, fᵧᵧ = −2, fₓᵧ = 0, so D = (2)(−2) − 0 = −4 < 0, confirming a saddle.

务必记住 D 需要在驻点处取值。经典例子：f(x,y) = x² − y²，其偏导 fₓ = 2x，fᵧ = −2y，驻点为 (0,0)。此时 fₓₓ = 2，fᵧᵧ = −2，fₓᵧ = 0，则 D = (2)(−2) − 0 = −4 < 0，确认为鞍点。

10. Optimisation with Constraints: Lagrange Multipliers | 带约束条件的优化：拉格朗日乘数法

To find the extreme values of f(x,y) subject to a constraint g(x,y) = 0, we introduce a Lagrange multiplier λ and solve the system ∇f = λ∇g together with g = 0. This yields ∂f/∂x = λ ∂g/∂x, ∂f/∂y = λ ∂g/∂y, and g(x,y) = 0.

为求函数 f(x,y) 在约束条件 g(x,y) = 0 下的极值，我们引入拉格朗日乘数 λ，并解方程组 ∇f = λ∇g 连同 g = 0。这给出 ∂f/∂x = λ ∂g/∂x，∂f/∂y = λ ∂g/∂y，以及 g(x,y) = 0。

For example, maximise f(x,y) = xy on the circle x² + y² = 1. Set g = x² + y² − 1. The equations are y = λ·2x, x = λ·2y, and x² + y² = 1. Solving typically gives candidate points (√2/2, √2/2), (−√2/2, −√2/2) etc., which can then be evaluated for maximum/minimum values.

例如，在圆 x² + y² = 1 上最大化 f(x,y) = xy。令 g = x² + y² − 1。方程组为 y = λ·2x，x = λ·2y，以及 x² + y² = 1。求解通常得到候选点 (√2/2, √2/2)、(−√2/2, −√2/2) 等，再代入判断是极大值还是极小值。

11. Common Applications and Examples | 常见应用与实例

Partial derivatives model rates of change in economics (marginal productivity, cross‑elasticity), thermodynamics (Maxwell relations), and geometry (tangent planes, normal lines). In exam questions, you may be asked to find the equation of a tangent plane: z − z₀ = fₓ(x₀,y₀)(x − x₀) + fᵧ(y₀,y₀)(y − y₀).

偏导数可用于描述经济学中的边际变化率（边际生产率、交叉弹性）、热力学中的麦克斯韦关系以及几何中的切平面与法线。考题可能要求写出切平面方程：z − z₀ = fₓ(x₀,y₀)(x − x₀) + fᵧ(x₀,y₀)(y − y₀)。

Another common task is error estimation: if a physical quantity Q depends on measured variables x and y with known errors, the maximum absolute error is approximately |∂Q/∂x| Δx + |∂Q/∂y| Δy. Understanding the total differential makes these calculations systematic.

另一常见题型是误差估计：若物理量 Q 依赖于带有已知误差的测量值 x 和 y，最大绝对误差近似为 |∂Q/∂x| Δx + |∂Q/∂y| Δy。理解全微分可以让这类计算更系统化。

12. Common Mistakes and Exam Tips | 常见错误与应试技巧

• Confusing ∂ and d: Use ∂ only for partial derivatives of multivariable functions; d is for ordinary derivatives. Writing dy/dx for a multivariable function will lose marks.
• Forgetting to treat other variables as constants: When finding ∂f/∂x, remember y, z, etc. are frozen. A quick mental check: substitute a number for y if unsure.
• Mixed derivative order confusion: Practice writing ∂²f/∂x∂y and fₓᵧ consistently. While Clairaut guarantees equality under continuity, exam questions may still test your understanding of order.
• Discriminant sign errors: Double-check D = fₓₓ fᵧᵧ − (fₓᵧ)². Many students mistakenly add or forget the square.

• 混淆 ∂ 与 d：∂ 仅用于多元函数的偏导数，d 是普通导数。在多元函数中写成 dy/dx 会失分。
• 忘记将其他变量视为常数：求 ∂f/∂x 时要记住 y、z 等是冻结的。如果不确定，可在心里代入一个具体数字帮助判断。
• 混合偏导求导顺序混淆：持续练习书写 ∂²f/∂x∂y 与 fₓᵧ。虽然克莱罗定理保证连续性下两者相等，考题仍可能考查对顺序的理解。
• 判别式符号错误：务必核对 D = fₓₓ fᵧᵧ − (fₓᵧ)²。很多学生错误地将其相加或忘记平方项。

Finally, always label your derivatives clearly and show the steps of evaluation at a point. When the test is inconclusive (D = 0), consider inspecting values along paths near the point or using second‑order Taylor expansion.

最后，务必清楚标注导数并在给定点计算时展示步骤。当判别式失效时（D = 0），考虑检查该点附近沿不同路径的取值，或使用二阶泰勒展开做进一步分析。

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Partial Differentiation Key Points | 偏微分考点精讲