Partial Differentiation: Key Concepts for IB & CIE Mathematics | IB CIE 数学：偏微分 考点精讲

📚 Partial Differentiation: Key Concepts for IB & CIE Mathematics | IB CIE 数学：偏微分 考点精讲

Partial differentiation is a fundamental tool in multivariable calculus. For students in IB and CIE Mathematics, mastering this topic is essential for solving problems involving rates of change in multiple dimensions, optimisation, and error estimation. This article covers all key ideas you need to know, presented in bilingual detail.

偏微分是多变量微积分的基本工具。对于学习IB与CIE数学的学生来说，掌握这一主题对于解决涉及多维变化率、最优化和误差估计的问题至关重要。本文涵盖你需要知道的所有关键知识点，并以双语形式详细讲解。

1. Introduction to Partial Derivatives | 偏导数介绍

When a function depends on more than one independent variable, such as z = f(x, y), we cannot simply speak of ‘the derivative’. Instead, we examine the rate of change with respect to one variable while holding the others constant. This leads to partial derivatives.

当一个函数依赖于多个自变量时，例如 z = f(x, y)，我们不能简单地谈论“导数”。相反，我们研究在保持其他变量不变的情况下，函数对某一个变量的变化率，这就引出了偏导数。

The partial derivative of f with respect to x, denoted ∂f/∂x or fₓ, is defined as the limit: ∂f/∂x = lim_{h→0} [f(x+h, y) – f(x, y)] / h, provided the limit exists. Geometrically, it gives the slope of the tangent line to the surface in the x-direction.

f 对 x 的偏导数，记作 ∂f/∂x 或 fₓ，定义为极限：∂f/∂x = lim_{h→0} [f(x+h, y) – f(x, y)] / h，假设该极限存在。从几何上看，它给出了曲面在 x 方向切线的斜率。

Similarly, the partial derivative with respect to y, ∂f/∂y or f_y, treats x as constant and measures the rate of change along the y-direction.

类似地，对 y 的偏导数 ∂f/∂y 或 f_y，将 x 视为常数，衡量沿 y 方向的变化率。

2. Notation and Basic Computation | 符号与基本计算

The two most common notations for partial derivatives are: ∂f/∂x and fₓ for the first partial with respect to x; ∂f/∂y and f_y for that with respect to y. We will use ∂z/∂x, ∂z/∂y, etc., for clarity.

偏导数最常见的两种记法是：对 x 的偏导数为 ∂f/∂x 和 fₓ；对 y 的为 ∂f/∂y 和 f_y。为清晰起见，我们将使用 ∂z/∂x, ∂z/∂y 等形式。

To compute ∂z/∂x, treat y as a constant and differentiate with respect to x using standard rules. Similarly, for ∂z/∂y, treat x as constant. All familiar differentiation rules – power, product, quotient, chain – apply.

计算 ∂z/∂x 时，将 y 视为常数，用通常的求导法则对 x 求导。类似地，对于 ∂z/∂y，将 x 视为常数。所有熟悉的求导法则——幂函数、乘积、商、链式——均适用。

Example: If z = 3x²y – 2xy³ + ln(x² + y²), find ∂z/∂x and ∂z/∂y.

例子：若 z = 3x²y – 2xy³ + ln(x² + y²)，求 ∂z/∂x 和 ∂z/∂y。

∂z/∂x = 6xy – 2y³ + (2x)/(x² + y²)

∂z/∂y = 3x² – 6xy² + (2y)/(x² + y²)

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

The second-order partial derivatives are obtained by differentiating the first partial derivatives again. Common notations: ∂²f/∂x² = (∂/∂x)(∂f/∂x), ∂²f/∂y² = (∂/∂y)(∂f/∂y). The mixed partial derivatives are ∂²f/∂x∂y and ∂²f/∂y∂x.

二阶偏导数是再次对一阶偏导数求导得到的。常用记法：∂²f/∂x² = (∂/∂x)(∂f/∂x)，∂²f/∂y² = (∂/∂y)(∂f/∂y)。混合偏导数为 ∂²f/∂x∂y 和 ∂²f/∂y∂x。

For most well-behaved functions encountered in exams, the mixed partial derivatives are equal: ∂²f/∂x∂y = ∂²f/∂y∂x. This is Clairaut’s theorem. It is an important symmetry property and can save time in computations.

对于考试中大多数性质良好的函数，混合偏导数是相等的：∂²f/∂x∂y = ∂²f/∂y∂x。这就是克莱罗定理。这是一个重要的对称性质，可以节省计算时间。

Example: f(x,y) = x³y² + e^(xy). Compute ∂²f/∂x², ∂²f/∂y² and ∂²f/∂x∂y.

例子：f(x,y) = x³y² + e^(xy)。计算 ∂²f/∂x², ∂²f/∂y² 和 ∂²f/∂x∂y。

∂f/∂x = 3x²y² + y e^(xy) ⇒ ∂²f/∂x² = 6xy² + y² e^(xy)

∂f/∂y = 2x³y + x e^(xy) ⇒ ∂²f/∂y² = 2x³ + x² e^(xy)

∂²f/∂x∂y = ∂/∂y(∂f/∂x) = 6x²y + e^(xy) + xy e^(xy)

4. The Chain Rule | 链式法则

If z = f(x, y) and both x and y are functions of a single variable t, then the total derivative with respect to t is given by: dz/dt = (∂z/∂x)(dx/dt) + (∂z/

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

更多咨询请联系16621398022（同微信）