Partial Differentiation Essentials for IB CCEA Maths | IB CCEA 数学：偏微分考点精讲

📚 Partial Differentiation Essentials for IB CCEA Maths | IB CCEA 数学：偏微分考点精讲

Partial differentiation extends the ideas of single‑variable calculus to functions of two or more variables. In IB CCEA Mathematics, mastering partial derivatives is essential for tackling multivariable optimisation, modelling rates of change in thermodynamics and economics, and understanding geometric concepts such as tangent planes. This article distils the core techniques, notation, and exam tips you need.

偏微分把单变量微积分的思想推广到两个或更多变量的函数中。在 IB CCEA 数学中，掌握偏导数对于处理多元优化问题、建立热力学和经济学中的变化率模型以及理解切平面等几何概念至关重要。本文浓缩了你需要的核心技巧、符号和应试要点。

1. What is Partial Differentiation? | 什么是偏微分？

A partial derivative of a function f(x, y) measures how the function changes when one variable changes while all other independent variables are held constant. For instance, ∂f/∂x gives the instantaneous rate of change of f with respect to x, treating y as a fixed parameter.

函数 f(x, y) 的偏导数衡量当某一个变量变化而所有其他自变量保持恒定时函数如何变化。例如，∂f/∂x 给出了 f 关于 x 的瞬时变化率，同时将 y 视为固定参数。

The process mirrors ordinary differentiation: differentiate with respect to the chosen variable and treat all other variables as constants. This simple rule underpins all partial differentiation techniques.

这个过程与普通求导相同：对所选的变量求导，同时把所有其他变量当作常数处理。这条简单的规则是所有偏微分技术的基础。

2. Notation and First‑Order Partial Derivatives | 符号与一阶偏导数

For a function z = f(x, y), the two first‑order partial derivatives are written as

对于函数 z = f(x, y)，两个一阶偏导数写作

∂f/∂x (or fₓ) and ∂f/∂y (or fᵧ)

Alternative notations include f₁(x, y) for the derivative with respect to the first argument, and f₂(x, y) for the second. In IB exams you may encounter any of these, so familiarity with all forms is vital.

其他记号包括 f₁(x, y) 表示对第一个自变量的导数，f₂(x, y) 表示对第二个自变量的导数。IB 考试中你可能会遇到其中任何一种，因此熟悉所有形式非常关键。

Example: For f(x, y) = 3x²y + e²ʸ sin x, we obtain ∂f/∂x = 6xy + e²ʸ cos x and ∂f/∂y = 3x² + 2e²ʸ sin x. Notice that when differentiating with respect to x, terms containing only y (like e²ʸ) are treated as constant multipliers.

示例：对于 f(x, y) = 3x²y + e²ʸ sin x，我们得到 ∂f/∂x = 6xy + e²ʸ cos x，∂f/∂y = 3x² + 2e²ʸ sin x。注意，对 x 求导时，仅包含 y 的项（如 e²ʸ）被视为常系数。

3. Geometric Interpretation | 几何意义

If z = f(x, y) represents a surface in three‑dimensional space, ∂f/∂x at a point (a, b) is the slope of the tangent line to the curve obtained by intersecting the surface with the plane y = b. Similarly, ∂f/∂y is the slope of the tangent line in the x = a plane.

若 z = f(x, y) 表示三维空间中的一个曲面，则在点 (a, b) 处 ∂f/∂x 是曲面与平面 y = b 相交所得曲线的切线斜率。类似地，∂f/∂y 是在 x = a 平面中切线的斜率。

These two tangent lines span the tangent plane to the surface at (a, b, f(a, b)). Understanding this geometry is crucial for visualising directional derivatives and gradients.

这两条切线张成曲面在 (a, b, f(a, b)) 处的切平面。理解这一几何关系对于直观把握方向导数和梯度至关重要。

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

Second‑order partial derivatives are obtained by differentiating first‑order derivatives. There are four possibilities for f(x, y):

二阶偏导数由一阶偏导数再次求导得到。对于 f(x, y)，有四种可能：

∂²f/∂x² : differentiate ∂f/∂x with respect to x again
∂²f/∂y² : differentiate ∂f/∂y with respect to y again
∂²f/∂y∂x : differentiate ∂f/∂x with respect to y
∂²f/∂x∂y : differentiate ∂f/∂y with respect to x

∂²f/∂x² : 再对 x 求 ∂f/∂x 的导数
∂²f/∂y² : 再对 y 求 ∂f/∂y 的导数
∂²f/∂y∂x : 对 y 求 ∂f/∂x 的导数
∂²f/∂x∂y : 对 x 求 ∂f/∂y 的导数

Clairaut’s Theorem states that if the second‑order mixed partials are continuous, then ∂²f/∂y∂x = ∂²f/∂x∂y. This symmetry often simplifies calculations and is regularly tested.

克莱罗定理指出，如果二阶混合偏导数连续，那么 ∂²f/∂y∂x = ∂²f/∂x∂y。这种对称性经常能够简化计算，也是常考的内容。

5. The Chain Rule for Multivariable Functions | 多元函数的链式法则

When z = f(x, y) and x, y are themselves functions of a single variable t (or several variables), the chain rule provides the total derivative. For x(t), y(t):

当 z = f(x, y) 且 x, y 本身是单个变量 t（或多个变量）的函数时，链式法则给出全导数。对于 x(t), y(t)：

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

If x and y depend on two variables u and v, the partial chain rule takes the form

如果 x 和 y 依赖于两个变量 u 和 v，偏导数的链式法则具有以下形式

∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)

and similarly for ∂z/∂v. These relations are indispensable in implicit differentiation and when changing coordinate systems (e.g., polar coordinates).

对于 ∂z/∂v 也有类似关系。这些关系在隐函数求导和变换坐标系（例如极坐标）时不可或缺。

6. Total Differential and Linear Approximation | 全微分与线性近似

The total differential dz approximates the change in z = f(x, y) when x and y change by small amounts dx and dy:

全微分 dz 近似表示当 x 和 y 分别发生微小变化 dx 和 dy 时 z = f(x, y) 的变化量：

dz = (∂f/∂x) dx + (∂f/∂y) dy

This leads to the linear (tangent‑plane) approximation near a point (a, b):

由此可得在点 (a, b) 附近的线性（切平面）近似：

f(x, y) ≈ f(a, b) + fₓ(a, b)(x − a) + fᵧ(a, b)(y − b)

This formula is the multivariable extension of the one‑variable tangent line approximation and is widely used in error estimation and sensitivity analysis.

该公式是单变量切线近似的多元推广，广泛用于误差估计和灵敏度分析。

7. Implicit Partial Differentiation | 隐函数偏微分

If an equation F(x, y, z) = 0 defines z implicitly as a function of x and y, we can find ∂z/∂x and ∂z/∂y without solving for z explicitly. Differentiate both sides with respect to x, treating y as constant and remembering that z depends on x:

如果方程 F(x, y, z) = 0 将 z 隐式地定义为 x 和 y 的函数，我们可以不必显式解出 z 而直接求出 ∂z/∂x 和 ∂z/∂y。对等式两边关于 x 求导，将 y 视为常数，并记住 z 依赖于 x：

∂F/∂x + (∂F/∂z)(∂z/∂x) = 0 ⇒ ∂z/∂x = − (∂F/∂x)/(∂F/∂z)

The same logic gives ∂z/∂y = − (∂F/∂y)/(∂F/∂z). This technique is highly testable, especially in conjunction with the chain rule.

同理可得 ∂z/∂y = − (∂F/∂y)/(∂F/∂z)。这一技巧极常考，尤其结合链式法则一同出现。

8. Directional Derivatives and the Gradient Vector | 方向导数与梯度向量

The gradient vector of f(x, y) is

f(x, y) 的梯度向量为

∇f = (∂f/∂x, ∂f/∂y)

The directional derivative of f at (x₀, y₀) in the direction of a unit vector u = (u₁, u₂) is the dot product of the gradient with u:

f 在 (x₀, y₀) 处沿单位向量 u = (u₁, u₂) 的方向导数等于梯度与 u 的点积：

D_uf = ∇f · u = fₓ u₁ + fᵧ u₂

The maximum rate of increase of f is |∇f| and occurs in the direction of ∇f. The gradient is always perpendicular to level curves (contours) of f.

f 的最大增长率是 |∇f|，并且沿 ∇f 的方向达到。梯度始终垂直于 f 的等值线（等高线）。

9. Tangent Planes and Normal Lines | 切平面与法线

For a surface given explicitly as z = f(x, y), the equation of the tangent plane at (a, b, f(a, b)) is

对于由显函数 z = f(x, y) 给出的曲面，在 (a, b, f(a, b)) 处的切平面方程为

z − f(a, b) = fₓ(a, b)(x − a) + fᵧ(a, b)(y − b)

A normal vector to this plane is n = ( −fₓ, −fᵧ, 1 ). For a surface given implicitly by F(x, y, z) = 0, the gradient ∇F = (Fₓ, Fᵧ, F₂) is a normal vector, and the tangent plane is

该平面的一个法向量是 n = ( −fₓ, −fᵧ, 1 )。对于由隐式方程 F(x, y, z) = 0 给出的曲面，梯度 ∇F = (Fₓ, Fᵧ, F₂) 是法向量，切平面方程为

Fₓ(a,b,c)(x − a) + Fᵧ(a,b,c)(y − b) + F₂(a,b,c)(z − c) = 0

Normal lines can be written immediately from these vectors.

从这些向量可以直接写出法线方程。

10. Applications to Optimisation (Critical Points) | 优化应用（临界点）

To find local maxima, minima, or saddle points of a function f(x, y) without constraints, set both first‑order partial derivatives to zero:

要寻找函数 f(x, y) 在无约束条件下的局部极大值、极小值或鞍点，令两个一阶偏导数同时为零：

fₓ(x, y) = 0 and fᵧ(x, y) = 0

Solutions are called critical (or stationary) points. Not all critical points are extrema; some are saddle points where the function increases in one direction and decreases in another.

其解称为临界点（或驻点）。并非所有临界点都是极值点；有些是鞍点，函数在某个方向上增加，而在另一个方向上减少。

11. Classifying Critical Points (Second Derivative Test) | 临界点分类（二阶导数检验）

At a critical point (a, b), compute the discriminant:

在临界点 (a, b) 处，计算判别式：

D = fₓₓ(a, b) fᵧᵧ(a, b) − [fₓᵧ(a, b)]²

The classification rules are summarised in the table:

分类规则总结于下表：

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

If D = 0, further investigation (e.g., along curves) is needed.

如果 D = 0，则需要进一步考察（例如沿曲线分析）。

12. Lagrange Multipliers (Constrained Optimisation) | 拉格朗日乘数法（约束优化）

To maximise or minimise f(x, y) subject to a constraint g(x, y) = 0, introduce the Lagrange function:

要对满足约束 g(x, y) = 0 的 f(x, y) 求最大值或最小值，引入拉格朗日函数：

L(x, y, λ) = f(x, y) − λ g(x, y)

Set the three partial derivatives equal to zero:

令三个偏导数为零：

∂L/∂x = 0, ∂L/∂y = 0, ∂L/∂λ = 0

Solving this system yields the candidates for constrained extrema. The method generalises to more variables and multiple constraints. It is a powerful tool in economic and physical applications.

求解该方程组得到约束极值的候选点。该方法可推广到更多变量和多个约束，是经济学和物理应用中的强有力工具。

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Partial Differentiation Essentials for IB CCEA Maths | IB CCEA 数学：偏微分 考点精讲