Partial Differentiation for IB and OCR Mathematics: Key Concepts

📚 Partial Differentiation for IB and OCR Mathematics: Key Concepts | 偏微分考点精讲

Partial differentiation extends the rules of single-variable calculus to functions of two or more variables. For IB and OCR further mathematics, mastering partial derivatives is crucial for analysing rates of change in multivariable contexts, classifying stationary points, and solving constrained optimisation problems. This article provides a comprehensive review of the essential techniques, from basic definitions to Lagrange multipliers, with paired English and Chinese explanations to reinforce understanding.

偏微分将单变量微积分的规则推广到二元或多元函数。对于 IB 和 OCR 进阶数学，掌握偏导数是分析多变量变化率、分类驻点以及求解约束优化问题的关键。本文从基本定义到拉格朗日乘数法，系统梳理核心技巧，并采用中英对照讲解以加深理解。

1. Definition and Notation of Partial Derivatives | 偏导数的定义与记号

For a function z = f(x, y), the partial derivative with respect to x at a point (a, b) is defined as the limit: ∂f/∂x = lim (h → 0) [f(a+h, b) – f(a,b)] / h, provided the limit exists. Geometrically, this measures the instantaneous rate of change of f in the x-direction while keeping y fixed. Common notations include f_x, ∂f/∂x, and ∂z/∂x.

对于函数 z = f(x, y)，关于 x 的偏导数在点 (a, b) 处定义为极限：∂f/∂x = lim (h → 0) [f(a+h, b) – f(a,b)] / h，假设该极限存在。几何上，它衡量在保持 y 不变时函数沿 x 方向的瞬时变化率。常用记号有 f_x、∂f/∂x 和 ∂z/∂x。

Similarly, the partial derivative with respect to y is obtained by treating x as a constant and differentiating with respect to y: ∂f/∂y = lim (k → 0) [f(a, b+k) – f(a,b)] / k. The process mirrors ordinary differentiation but requires careful attention to which variables are held constant.

类似地，关于 y 的偏导数通过将 x 视为常数并对 y 求导得到：∂f/∂y = lim (k → 0) [f(a, b+k) – f(a,b)] / k。这个过程与普通求导相似，但需要特别注意哪些变量被当作常数处理。

2. Computing Partial Derivatives | 计算偏导数

To find f_x for f(x, y) = x³y + sin(xy), treat y as a constant and differentiate with respect to x: f_x = 3x²y + y·cos(xy). For f_y, treat x as constant: f_y = x³ + x·cos(xy). Standard rules—product, quotient, chain—apply term by term.

对于 f(x, y) = x³y + sin(xy)，求 f_x 时将 y 视为常数并对 x 求导：f_x = 3x²y + y·cos(xy)。求 f_y 时将 x 视为常数：f_y = x³ + x·cos(xy)。乘积法则、商法则、链式法则等标准规则逐项适用。

When dealing with more complex functions, always identify which variable is being differentiated and consistently treat all others as constants. For instance, if z = ln(x² + y²), then ∂z/∂x = (2x) / (x² + y²) and ∂z/∂y = (2y) / (x² + y²). Note the symmetry when the function is symmetric in x and y.

处理更复杂的函数时，务必明确对哪一个变量求导，并始终将其他变量视为常数。例如，若 z = ln(x² + y²)，则 ∂z/∂x = (2x) / (x² + y²)，∂z/∂y = (2y) / (x² + y²)。注意当函数关于 x 和 y 对称时，其结果也体现了对称性。

3. Higher-Order and Mixed Partial Derivatives | 高阶偏导数与混合偏导数

Second-order partial derivatives are obtained by differentiating first-order partials. The notation f_xx = ∂²f/∂x² means differentiate f twice with respect to x; f_yy = ∂²f/∂y² means twice with respect to y. Mixed partials f_xy = ∂²f/∂y∂x, differentiate first with respect to x then y. Under typical smoothness conditions, f_xy = f_yx (Clairaut’s theorem).

二阶偏导数由一阶偏导数再次求导得到。记号 f_xx = ∂²f/∂x² 表示对 x 求导两次；f_yy = ∂²f/∂y² 表示对 y 求导两次。混合偏导数 f_xy = ∂²f/∂y∂x 表示先对 x 求导再对 y 求导。在通常的光滑性条件下，f_xy = f_yx（克莱罗定理）。

Example: For f(x, y) = x²e^y + y³, we have f_x = 2x e^y, f_xx = 2e^y, f_y = x²e^y + 3y², f_yy = x²e^y + 6y. The mixed derivative f_xy = ∂/∂y (2x e^y) = 2x e^y, and f_yx = ∂/∂x (x²e^y + 3y²) = 2x e^y, confirming equality.

例如：对于 f(x, y) = x²e^y + y³，我们有 f_x = 2x e^y，f_xx = 2e^y，f_y = x²e^y + 3y²，f_yy = x²e^y + 6y。混合偏导数 f_xy = ∂/∂y (2x e^y) = 2x e^y，f_yx = ∂/∂x (x²e^y + 3y²) = 2x e^y，验证了相等性。

4. Total Differentials and Approximation | 全微分与近似计算

If z = f(x, y) has continuous partial derivatives near (a, b), the total differential is dz = f_x(a,b) dx + f_y(a,b) dy. This linear approximation gives the estimated change in z for small changes dx and dy. It generalises the one-variable differential dy = f ‘(x)dx to higher dimensions.

若 z = f(x, y) 在 (a, b) 附近有连续偏导数，全微分为 dz = f_x(a,b) dx + f_y(a,b) dy。该线性逼近给出了当 x 和 y 有微小变化 dx、dy 时 z 的近似变化量。它是单变量微分 dy = f ‘(x)dx 向高维的推广。

In practice, we use the linearisation L(x, y) = f(a, b) + f_x(a,b)(x–a) + f_y(a,b)(y–b) to estimate function values near (a, b). For example, if f(x, y) = √(x²+y²), at (3,4) we have f_x = 3/5, f_y = 4/5. The linear approximation near (3,4) is L(x, y) = 5 + (3/5)(x–3) + (4/5)(y–4).

应用中，我们使用线性化 L(x, y) = f(a, b) + f_x(a,b)(x–a) + f_y(a,b)(y–b) 来估计点 (a, b) 附近的函数值。例如，若 f(x, y) = √(x²+y²)，在 (3,4) 处有 f_x = 3/5, f_y = 4/5。在 (3,4) 附近的线性逼近为 L(x, y) = 5 + (3/5)(x–3) + (4/5)(y–4)。

Error propagation in measurements often uses total differentials: if a quantity is calculated from measurements with small uncertainties, the maximum absolute error is approximated by |f_x Δx| + |f_y Δy|, where Δx, Δy are measurement errors.

测量中的误差传播常使用全微分：如果一个量由带有微小不确定度的测量值计算得出，最大绝对误差可用 |f_x Δx| + |f_y Δy| 近似，其中 Δx、Δy 为测量误差。

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

When z = f(x, y) and both x and y depend on a single parameter t, the total derivative is dz/dt = (∂f/∂x) (dx/dt) + (∂f/∂y) (dy/dt). If x and y themselves depend on two variables u and v, partial derivatives follow: ∂z/∂u = (∂f/∂x) (∂x/∂u) + (∂f/∂y) (∂y/∂u). The pattern depends on the dependence structure.

当 z = f(x, y)，且 x、y 均依赖于单一参数 t，则全导数为 dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)。若 x 和 y 本身依赖两个变量 u、v，则偏导数遵循：∂z/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u)。具体形式取决于依赖结构。

Example: Let f(x, y) = x² + y², with x = e^t and y = sin(t). Then dz/dt = 2x·e^t + 2y·cos(t) = 2e^2t + 2 sin(t) cos(t). Instead of substituting first, the chain rule offers a more efficient path, especially for complex expressions.

例如：设 f(x, y) = x² + y²，其中 x = e^t、y = sin(t)。则 dz/dt = 2x·e^t + 2y·cos(t) = 2e^2t + 2 sin(t) cos(t)。链式法则避免了先代入再求导，对复杂表达式尤其高效。

A common pitfall is mixing partial and ordinary derivatives. Use ∂ only when differentiating a multivariable function with respect to one of its independent variables, and d when the function ultimately depends on a single variable.

常见的误区是混淆偏导与常导。仅当对多元函数的某个自变量求导时使用 ∂，当函数最终依赖于单一变量时使用 d。

6. Implicit Differentiation with Partial Derivatives | 隐函数的偏导数求导

For an equation F(x, y) = 0 that implicitly defines y as a function of x, implicit differentiation using partials gives dy/dx = – F_x / F_y, provided F_y ≠ 0. This formula is derived by viewing F as a function of two variables and applying the chain rule to F(x, y(x)) = 0.

对于方程 F(x, y) = 0 隐式地将 y 定义为 x 的函数，使用偏导数的隐函数求导给出 dy/dx = – F_x / F_y，前提是 F_y ≠ 0。此公式通过将 F 视为二元函数并对 F(x, y(x)) = 0 应用链式法则导出。

For a surface defined implicitly by F(x, y, z) = 0, we can find partial derivatives of z with respect to x and y: ∂z/∂x = – F_x / F_z, and ∂z/∂y = – F_y / F_z. This is particularly useful when solving for z explicitly is difficult.

对于由 F(x, y, z) = 0 隐式定义的曲面，可求 z 关于 x 和 y 的偏导数：∂z/∂x = – F_x / F_z，∂z/∂y = – F_y / F_z。当显式解出 z 比较困难时，该方法尤为有用。

Example: The curve x²y + y³ = 2 can be written as F(x,y)= x²y + y³ – 2 = 0. Then dy/dx = – (2xy) / (x²+3y²). At the point (1,1), the slope is –2/4 = –1/2.

例如：曲线 x²y + y³ = 2 可写成 F(x,y)= x²y + y³ – 2 = 0。则 dy/dx = – (2xy) / (x²+3y²)。在点 (1,1) 处，斜率为 –2/4 = –1/2。

7. Classifying Stationary Points of Two-Variable Functions | 二元函数驻点的分类

A stationary point of f(x, y) occurs where both first-order partials are zero: f_x = 0 and f_y = 0. To classify, compute the second partials and the discriminant D = f_xx f_yy – (f_xy)² evaluated at the stationary point.

f(x, y) 的驻点出现在两个一阶偏导数均为零处：f_x = 0 且 f_y = 0。分类时，计算二阶偏导数以及判别式 D = f_xx f_yy – (f_xy)²，在驻点处取值。

The classification rule:

If D > 0 and f_xx > 0, the point is a local minimum.
If D > 0 and f_xx < 0, it is a local maximum.
If D < 0, the point is a saddle point.
If D = 0, the test is inconclusive.

分类规则：

若 D > 0 且 f_xx > 0，该点为局部极小值点。
若 D > 0 且 f_xx < 0，该点为局部极大值点。
若 D < 0，该点为鞍点。
若 D = 0，则无法判定。

Example: Find and classify stationary points of f(x, y) = x² + y² + 4x – 6y + 10. Setting f_x = 2x + 4 = 0 gives x = –2; f_y = 2y – 6 = 0 gives y = 3. The second partials: f_xx = 2, f_yy = 2, f_xy = 0. Hence D = 2×2 – 0² = 4 > 0 and f_xx > 0, so (–2,3) is a local minimum.

示例：求 f(x, y) = x² + y² + 4x – 6y + 10 的驻点并分类。令 f_x = 2x + 4 = 0 得 x = –2；f_y = 2y – 6 = 0 得 y = 3。二阶偏导：f_xx = 2，f_yy = 2，f_xy = 0。因此 D = 2×2 – 0² = 4 > 0 且 f_xx > 0，故 (–2,3) 为局部极小值点。

In IB and OCR further maths, you may need to justify using the discriminant, showing the second derivatives clearly. Always check if the function has any constraints; otherwise use Lagrange multipliers.

在 IB 和 OCR 进阶数学中，你需要使用判别式进行论证，清晰展示二阶导数。务必检查函数是否带有约束条件；否则使用拉格朗日乘数法。

8. Lagrange Multipliers for Constrained Optimization | 拉格朗日乘数法求解约束优化

To find extrema of f(x, y) subject to a constraint g(x, y) = k, we set up the Lagrangian L(x, y, λ) = f(x, y) + λ (k – g(x, y)). Then solve the system: ∂L/∂x = 0, ∂L/∂y = 0, and ∂L/∂λ = 0 (which recovers the constraint). The variable λ is the Lagrange multiplier.

为求 f(x, y) 在约束 g(x, y) = k 下的极值，构造拉格朗日函数 L(x, y, λ) = f(x, y) + λ (k – g(x, y))。然后求解方程组：∂L/∂x = 0，∂L/∂y = 0，以及 ∂L/∂λ = 0（即恢复约束条件）。变量 λ 称为拉格朗日乘数。

Example: Maximise f(x, y) = xy subject to x + y = 10. Set L = xy + λ(10 – x – y). Then ∂L/∂x = y – λ = 0, ∂L/∂y = x – λ = 0, and ∂L/∂λ = 10 – x – y = 0. From the first two, x = y = λ, substitute into the third: 10 – 2x = 0 ⇒ x = 5, y = 5, λ = 5. The maximum value is 25.

例如：在约束 x + y = 10 下最大化 f(x, y) = xy。设 L = xy + λ(10 – x – y)。则 ∂L/∂x = y – λ = 0，∂L/∂y = x – λ = 0，∂L/∂λ = 10 – x – y = 0。由前两式得 x = y = λ，代入第三式：10 – 2x = 0 ⇒ x = 5, y = 5, λ = 5。最大值为 25。

For constraints of the form g(x,y) = k, sometimes it is easier to solve directly by substitution, but Lagrange multipliers are systematic for more complex or multiple constraints. In IB exams, questions may involve geometric contexts like optimising area or volume given a fixed perimeter or surface area.

对于约束 g(x,y) = k，有时直接代入求解更简单，但拉格朗日乘数在处理较复杂或多重约束时更为系统。在 IB 考试中，题目可能涉及几何情境，如在固定周长或表面积下优化面积或体积。

The method extends to three variables and more constraints: for f(x,y,z) subject to g(x,y,z)=k, L = f + λ(k – g), solving four equations. Interpret λ as the rate of change of the optimum value with respect to the constraint constant.

该方法可推广到三个变量及更多约束：对 f(x,y,z) 在 g(x,y,z)=k 下，L = f + λ(k – g)，求解四个方程。λ 的经济学解释为最优值随约束常数变化的速率。

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Partial Differentiation for IB and OCR Mathematics: Key Concepts | 偏微分考点精讲