Partial Differentiation for AS Mathematics | AS数学偏微分考点精讲

📚 Partial Differentiation for AS Mathematics | AS数学偏微分考点精讲

Partial differentiation is a key topic in AS-level mathematics, particularly in further pure modules. It extends the concept of ordinary differentiation to functions of two or more variables, enabling us to study rates of change along specific directions. Mastering partial derivatives is essential for tackling problems in optimisation, geometry, and many applied contexts. This guide systematically covers the core concepts, techniques, and exam-style applications that you need.

偏微分是AS数学（尤其是进阶纯数模块）中的一个重要考点。它将一元函数微分的概念推广到多元函数，使我们能够研究沿特定方向的变化率。掌握偏导数对于解决优化问题、几何问题以及许多应用场景至关重要。本指南将系统梳理核心概念、计算技巧和考试常见题型，助你轻松应对。

---

1. Functions of Several Variables | 多元函数简介

pairs (x, y). Common examples include f(x, y) = x² + y² and g(x, y) = sin(xy). Understanding the domain and the graphical representation is the first step.

在普通微积分中，我们处理单变量函数，例如 f(x)。而在偏微分中，我们考虑两个或多个自变量的函数，例如 f(x, y) 或 z = f(x, y)。这类函数定义了三维空间中的曲面。其定义域是有序对 (x, y) 的集合。常见的例子如 f(x, y) = x² + y² 和 g(x, y) = sin(xy)。理解定义域及其图形表示是学习的第一步。

---

2. Partial Derivatives: Definition and Notation | 偏导数的定义与记号

The partial derivative of f(x, y) with respect to x, denoted ∂f/∂x or f_x d by treating y as a constant and differentiating with respect to x. Similarly, ∂f/∂y or f_y treats x as constant. Formally, ∂f/∂x = lim_h→0 [f(x+h, y) - f(x, y)] / h. We use the curly '∂' symbol to distinguish from ordinary derivatives.

函数 f(x, y) 关于 x 的偏导数记为 ∂f/∂x 或 f_x，求法是将 y 视为常数，然后对 x 求导。类似地，∂f/∂y 或 f_y 则是把 x 视为常数对 y 求导。正式定义为 ∂f/∂x = lim_h→0 [f(x+h, y) - f(x, y)] / h。我们用卷曲的 ∂ 符号来区别于普通导数。

For example, if f(x, y) = x²y + sin y, then:

例如，若 f(x, y) = x²y + sin y，则：

∂f/∂x = 2xy, ∂f/∂y = x² + cos y

---

3. Computing First-Order Partial Derivatives | 一阶偏导数的计算

Calculating first-order partial derivatives follows standard differentiation rules while keeping the other variable constant. Let’s illustrate with common function types.

计算一阶偏导数时遵循标准求导法则，同时将其他变量视为常数。下面通过常见函数类型加以说明。

Polynomial and power functions: For f(x, y) = 3x⁴y² + 2x − y³, we obtain ∂f/∂x = 12x³y² + 2 and ∂f/∂y = 6x⁴y − 3y².

多项式与幂函数：对于 f(x, y) = 3x⁴y² + 2x − y³，可得 ∂f/∂x = 12x³y² + 2，∂f/∂y = 6x⁴y − 3y²。

Exponential and trigonometric functions: If g(x, y) = e^xy sin(x + y), then ∂g/∂x = y e^xy sin(x + y) + e^xy cos(x + y), and ∂g/∂y = x e^xy sin(x + y) + e^xy cos(x + y).

指数与三角函数：若 g(x, y) = e^xy sin(x + y)，则 ∂g/∂x = y e^xy sin(x + y) + e^xy cos(x + y)，∂g/∂y = x e^xy sin(x + y) + e^xy cos(x + y)。

Logarithmic functions: , y) = ln(x² + y²), partial derivatives are ∂h/∂x = 2x/(x² + y²) and ∂h/∂y = 2y/(x² + y²).

对数函数：对于 h(x, y) = ln(x² + y²)，偏导数为 ∂h/∂x = 2x/(x² + y²)，∂h/∂y = 2y/(x² + y²)。

---

4. Geometric Interpretation | 几何意义

ese slopes describe the steepness of the surface in the directions of the coordinate axes.

对于曲面 z = f(x, y)，在点 (a, b) 处的偏导数 ∂f/∂x 表示曲面与平面 y = b 相交所得曲线的切线斜率。同理，∂f/∂y 给出在平面 x = a 内的切线斜率。这些斜率描述了曲面沿坐标轴方向的陡峭程度。

This interpretation is vital for understanding directional changes and will later connect to gradient vectors and optimisation in multivariable calculus.

这一几何解释对于理解方向性变化至关重要，并将与多元微积分中的梯度向量和优化问题联系起来。

---

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

ives again to obtain second-order partial derivatives. For z = f(x, y), the notations are:

我们可以再次对偏导数求导以获得二阶偏导数。对于 z = f(x, y)，记号如下：

∂²f/∂x² = f_xx, ∂²f/∂y² = f_yy, ∂²f/∂x∂y = f_xy, ∂²f/∂y∂x = f_yx

Clairaut's theorem states that if the second partial derivatives are continuous, then the mixed partials are equal: f_xy = f_yx. This symmetry often simplifies calculations in exam problems.

克莱罗定理指出，若二阶偏导数连续，则混合偏导数相等：f_xy = f_yx。这种对称性经常能在考试中简化计算。

Example: For f(x, y) = x³y² + e^xy, we obtain f_xx = 6xy² + e^xy, f_yy = 2x³, and f_xy = 6x²y + e^x, which matches f_yx.

示例：对于 f(x, y) = x³y² + e^xy，可得 f_xx = 6xy² + e^xy，f_yy = 2x³，以及 f_xy = 6x²y + e^x，与 f_yx 一致。

---

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

ctice, we use Δz ≈ (∂z/∂x) Δx + (∂z/∂y) Δy to estimate small changes or percentage errors.

函数 z = f(x, y) 的全微分定义为 dz = (∂z/∂x) dx + (∂z/∂y) dy。当 x 和 y 发生微小变化 Δx 和 Δy 时，它给出了 z 变化量的线性近似。实际应用中，我们使用 Δz ≈ (∂z/∂x) Δx + (∂z/∂y) Δy 来估算微小变化或百分误差。

oint (2, 3), then Δz ≈ (2xy)·0.1 + (x²)·(−0.05) = 2·2·3·0.1 + 4·(−0.05) = 1.2 − 0.2 = 1.0.

例如，若 z = x²y，在点 (2, 3) 处 x 增加 0.1，y 减少 0.05，则 Δz ≈ (2xy)·0.1 + (x²)·(−0.05) = 2·2·3·0.1 + 4·(−0.05) = 1.2 − 0.2 = 1.0。

---

广告 / Sponsored

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

functions of a single variable t, the derivative of z with respect to t is given by the chain rule:

当 z = f(x, y) 且 x, y 本身是关于单变量 t 的函数时，z 对 t 的导数由链式法则给出：

dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)

∂y/∂u). Remember to substitute the partial derivatives of the original function correctly.

如果 x 和 y 依赖于两个变量 u 和 v，则有更一般的形式。例如，若 x = u²v，y = uv²，则 ∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)。注意要正确代入原函数的偏导数。

This rule is essential when handling related rates or when converting between coordinate systems in applied problems.

在处理相关变化率或应用问题中坐标系转换时，这一法则必不可少。

---

8. Implicit Differentiation with Partial Derivatives | 隐函数偏导数

hat defines y implicitly as a function of x, the derivative dy/dx can be found using partial derivatives:

对于形如 F(x, y) = 0 的方程，它隐式地将 y 定义为 x 的函数，导数 dy/dx 可通过偏导数求得：

dy/dx = −F_x / F_y, provided F_y ≠ 0

for an implicit surface F(x, y, z) = 0, we can obtain partial derivatives like ∂z/∂x = −F_x/F_z. This technique is widely used in coordinate geometry and thermodynamics problems.

类似地，对于隐式曲面 F(x, y, z) = 0，可求得 ∂z/∂x = −F_x/F_z。这一技巧广泛应用于坐标几何和热力学问题中。

at as F(x, y, z) = x² + y² + z² − 4 = 0. Then ∂z/∂x = −(2x)/(2z) = −x/z, and ∂z/∂y = −y/z.

示例：若 x² + y² + z² = 4，视为 F(x, y, z) = x² + y² + z² − 4 = 0。则 ∂z/∂x = −(2x)/(2z) = −x/z，∂z/∂y = −y/z。

---

9. Stationary Points of Two-Variable Functions | 二元函数的驻点

equations yields the coordinates of critical points (a, b). These points are candidates for local maxima, minima, or saddle points.

函数 f(x, y) 的驻点是使两个一阶偏导数同时为零的点：∂f/∂x = 0 且 ∂f/∂y = 0。求解该方程组可得到临界点坐标 (a, b)。这些点可能是局部极大值、极小值或鞍点的候选点。

To find them, compute f_x and f_y, set them equal to zero, and solve the simultaneous equations. Often factoring helps to identify multiple solutions.

为求驻点，计算 f_x 和 f_y 并令其为零，然后解联立方程。因式分解通常有助于找到多个解。

Example: f(x, y) = x³ + 3xy² − 3x² − 3y² + 4. Then f_x = 3x² + 3y² − 6x = 0, f_y = 6xy − 6y = 6y(x − 1) = 0. Solve to obtain points (0,0), (2,0), (1,1), (1,-1).

示例：f(x, y) = x³ + 3xy² − 3x² − 3y² + 4。则 f_x = 3x² + 3y² − 6x = 0，f_y = 6xy − 6y = 6y(x − 1) = 0。求解得到点 (0,0), (2,0), (1,1), (1,-1)。

---

10. Classifying Stationary Points | 驻点类型的判别

To classify the stationary points, we use the second derivative test. Compute the second partial derivatives f_xx, f_yy, f_xy at the critical point, and evaluate the discriminant:

为判别驻点类型，我们使用二阶导数检验。在临界点处计算二阶偏导数 f_xx, f_yy, f_xy，并计算判别式：

D = f_xx f_yy − (f_xy)²

The classification rules are summarised in the table below.

分类规则总结于下表。

Condition	Conclusion
D > 0 and fxx < 0	Local maximum
D > 0 and fxx > 0	Local minimum
D < 0	Saddle point
D = 0	Test inconclusive

中文对应：当 D > 0 且 f_xx < 0 时为局部极大值；D > 0 且 f_xx > 0 时为局部极小值；D < 0 时为鞍点；D = 0 时检验失效。

Apply the test to the earlier example at (2,0): f_xx = 6x − 6 = 6, f_yy = 6x − 6 = 6, f_xy = 6y = 0, so D = 36 > 0 and f_xx > 0, indicating a local minimum.

将检验应用于之前的例子 (2,0)：f_xx = 6x − 6 = 6, f_yy = 6x − 6 = 6, f_xy = 6y = 0，故 D = 36 > 0 且 f_xx > 0，表明为局部极小值。

---

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

To excel in AS partial differentiation questions, keep these tips in mind and avoid typical pitfalls.

想在AS偏微分题目中取得高分，请牢记以下技巧并避免常见错误。

• Always specify which variable is held constant. A common mistake is forgetting to treat the other variable as a constant, leading to incorrect derivatives.
  务必明确固定哪个变量。忘记将其他变量视为常数是常见错误，会导致导数错误。
• Check mixed partial derivatives for equality when verifying second derivatives; this can catch arithmetic mistakes.
  利用混合偏导数的相等性来检验二阶导数，能够发现计算错误。
• When computing the discriminant D, be careful with signs – f_xy is squared, so negative values become positive, but miscalculating f_xy itself is a frequent error.
  计算判别式 D 时注意符号——f_xy 会被平方，因此负值变为正，但 f_xy 本身的计算错误十分常见。
• For stationary points, solve f_x = 0 and f_y = 0 systematically – factorisation often reveals all solutions; missing a solution loses marks.
  求驻点时，系统求解 f_x=0 和 f_y=0——因式分解常能揭示所有解；遗漏解会失分。
• Use the total differential for approximations only when changes are small; otherwise the linear approximation may be inaccurate.
  仅在变化微小时使用全微分进行近似；否则线性近似可能不准确。
• In implicit differentiation, remember the negative sign in dy/dx = −F_x/F_y. Omitting the minus is a classic slip.
  隐函数求导时，牢记 dy/dx = −F_x/F_y 中的负号。漏掉负号是经典失误。

Published by TutorHao | AS Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）