📚 A-Level WJEC Maths: Partial Differentiation Exam-Focused Revision | A-Level WJEC 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 WJEC A-Level Mathematics, this topic appears in the pure or further pure modules and is a key skill for tackling optimisation problems, finding rates of change in multivariable contexts, and understanding the geometry of surfaces. This article covers all essential techniques, from first-order partial derivatives to classifying stationary points using the Hessian determinant, with clear explanations and examination tips.
偏微分将单变量微积分的概念推广到两个或更多变量的函数中。在 WJEC A-Level 数学中，本主题出现在纯数学或进阶纯数模块里，是解决多变量优化问题、求多变量变化率以及理解曲面几何的关键技能。本文涵盖从一阶偏导数到利用海塞判别式对驻点进行分类的全部核心技巧，并配有清晰的解释与应试建议。
1. Functions of Two Variables | 二元函数
A function of two variables, such as f(x, y) or z = f(x, y), assigns a unique output z for each ordered pair (x, y) in a domain of the xy-plane. The graph of such a function is a surface in three-dimensional space, and partial derivatives describe how the surface slopes in directions parallel to the coordinate axes.
二元函数，例如 f(x, y) 或 z = f(x, y)，对 xy 平面定义域内的每一个有序对 (x, y) 都指定唯一的输出 z。这种函数的图像是三维空间中的一张曲面，而偏导数描述该曲面沿坐标轴方向的倾斜程度。
To work with these functions, we often need to evaluate expressions like f(2, -1) by substituting the numbers directly, or we may be asked to find domains where a formula is defined – for instance, ensuring the argument of a square root is non-negative or a denominator is not zero.
处理这类函数时，我们经常需要直接代入数值计算如 f(2, -1)，也可能被要求找出公式有定义的定义域——例如保证平方根中的量非负，或分母不为零。
2. First-Order Partial Derivatives | 一阶偏导数
The first-order partial derivative with respect to x, denoted ∂f/∂x or fx, is obtained by differentiating f(x, y) with respect to x while treating y as a constant. Similarly, ∂f/∂y or fy is found by differentiating with respect to y, holding x constant.
关于 x 的一阶偏导数，记作 ∂f/∂x 或 fx，是将 y 视为常数而对 x 求导得到的。类似地，∂f/∂y 或 fy 是在 x 视为常数的条件下对 y 求导。
For example, if f(x, y) = 3x²y + 2xy³ – y², then fx = 6xy + 2y³ (treat y as constant, so derivative of 3x²y is 6xy, derivative of 2xy³ is 2y³, and derivative of -y² is 0). And fy = 3x² + 6xy² – 2y (treat x as constant).
例如，若 f(x, y) = 3x²y + 2xy³ – y²，则 fx = 6xy + 2y³（将 y 视为常数，3x²y 的导数是 6xy，2xy³ 的导数是 2y³，-y² 的导数是 0）。fy = 3x² + 6xy² – 2y（x 视为常数）。
In WJEC exam questions, you may be given a composite function or one involving exponentials, logarithms or trigonometric terms. The same rule applies: hold one variable fixed and differentiate normally.
在 WJEC 考题中，你可能会遇到复合函数，或包含指数、对数、三角项的函数。规则完全相同：固定一个变量，正常求导。
3. Second-Order Partial Derivatives | 二阶偏导数
The second-order partial derivatives are obtained by differentiating the first-order derivatives. There are four possible second-order derivatives: ∂²f/∂x² (fxx), ∂²f/∂y² (fyy), ∂²f/∂x∂y (fxy), and ∂²f/∂y∂x (fyx).
二阶偏导数是将一阶偏导数再次求导得到。共有四种可能的二阶偏导数：∂²f/∂x² (fxx)、∂²f/∂y² (fyy)、∂²f/∂x∂y (fxy) 和 ∂²f/∂y∂x (fyx)。
To find fxx, differentiate fx with respect to x again, treating y as constant. To find fyy, differentiate fy with respect to y again, holding x constant. For the mixed derivative fxy, differentiate fx with respect to y, or fy with respect to x – the result is usually the same.
求 fxx 时，将 fx 再次对 x 求导，y 仍视为常数。求 fyy 时，将 fy 再次对 y 求导，x 视为常数。对于混合导数 fxy，可将 fx 对 y 求导，或将 fy 对 x 求导——结果通常相同。
Example: from the previous f(x, y) = 3x²y + 2xy³ – y², we have fxx = 6y, fyy = 12xy – 2, and fxy = fyx = 6x + 6y².
例如：从上例 f(x, y) = 3x²y + 2xy³ – y²，有 fxx = 6y，fyy = 12xy – 2，以及 fxy = fyx = 6x + 6y²。
4. Mixed Partial Derivatives and Clairaut’s Theorem | 混合偏导数与克莱罗定理
Clairaut’s theorem states that if the mixed partial derivatives fxy and fyx are continuous on an open set, then they are equal. For almost all functions encountered at A-Level, this condition holds, so we can safely compute the simpler mixed derivative.
克莱罗定理指出，如果混合偏导数 fxy 和 fyx 在某个开集上连续，则它们相等。对于A-Level 遇到的几乎所有函数，这一条件都成立，因此我们可以安全地计算较简单的那个混合导数。
This symmetry is extremely useful when one order of differentiation is algebraically easier than the other. For instance, if f(x, y) = eˣ sin y, then fx = eˣ sin y, and differentiating with respect to y gives fxy = eˣ cos y. The reverse order yields the same result but might involve more steps.
当一种求导顺序在代数上比另一种更简单时，这种对称性极为有用。例如，若 f(x, y) = eˣ sin y，则 fx = eˣ sin y，再对 y 求导得 fxy = eˣ cos y。 反向顺序得到相同结果，但可能步骤更多。
In WJEC exam questions, you may be asked to verify that fxy = fyx for a particular function, so be prepared to compute both and confirm equality.
在 WJEC 试题中，可能会要求验证某个函数的 fxy = fyx，因此要准备好计算两者并确认相等。
5. Chain Rule for Partial Derivatives | 偏导数的链式法则
If z = f(x, y) and x and y are themselves functions of a single variable t, then the total derivative dz/dt is given by the chain rule: dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt). This allows us to find how z changes with respect to t when x and y are linked through a parameter.
若 z = f(x, y)，且 x 和 y 本身又是单变量 t 的函数，则全导数 dz/dt 由链式法则给出：dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)。这使我们能够在 x 和 y 通过参数相联系时，求出 z 关于 t 的变化率。
If instead x and y are functions of two variables, say u and v, then we have the multivariate chain rule: ∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u) and similarly for ∂z/∂v. This is a favourite topic in WJEC papers, often tested with functions like z = x²y, x = u + v, y = uv.
如果 x 和 y 是两个变量（例如 u 和 v）的函数，则有多元链式法则：∂z/∂u = (∂z/∂x)(∂x/∂u) + (∂z/∂y)(∂y/∂u)，∂z/∂v 类似。这是 WJEC 试卷中的常见考点，经常用诸如 z = x²y，x = u + v，y = uv 的函数进行考查。
Always write the chain rule formula first, then substitute and simplify. Pay careful attention to which variables are being held constant at each stage.
始终先写出链式法则公式，再代入和化简。注意每个阶段中哪些变量被视为常数。
6. Implicit Partial Differentiation | 隐函数偏微分
An equation of the form F(x, y, z) = 0 may define z implicitly as a function of x and y. To find ∂z/∂x, differentiate the whole equation with respect to x, treating y as constant and remembering that z is a function of x, so we use the chain rule on terms involving z. Then solve for ∂z/∂x.
形如 F(x, y, z) = 0 的方程可能隐含地将 z 定义为 x 和 y 的函数。为求 ∂z/∂x，将整个方程对 x 求导，将 y 视为常数，并记住 z 是 x 的函数，因此对包含 z 的项使用链式法则。然后解出 ∂z/∂x。
For example, given x² + y² + z² = 1, differentiate with respect to x: 2x + 0 + 2z·(∂z/∂x) = 0, so ∂z/∂x = -x/z. Similarly, ∂z/∂y = -y/z. This technique extends to more complicated expressions and often appears alongside tangent plane problems.
例如，已知 x² + y² + z² = 1，对 x 求导：2x + 0 + 2z·(∂z/∂x) = 0，所以 ∂z/∂x = -x/z。类似地，∂z/∂y = -y/z。这一技巧可推广到更复杂的表达式，并常与切平面问题一同出现。
WJEC questions sometimes ask for the partial derivatives of implicit functions to then find the equation of a tangent plane or the normal line to a surface. Master implicit differentiation early in your revision.
WJEC 的题目有时会先要求求出隐函数的偏导数，再求曲面的切平面方程或法线。复习时要尽早掌握隐函数微分法。
7. Stationary Points of Functions of Two Variables | 二元函数的驻点
A stationary point of z = f(x, y) occurs where both first-order partial derivatives are simultaneously zero: fx = 0 and fy = 0. Solving these equations simultaneously yields the critical points (x₀, y₀).
二元函数 z = f(x, y) 的驻点出现在两个一阶偏导数同时为零的位置：fx = 0 且 fy = 0。联立解这两个方程就得到临界点 (x₀, y₀)。
To find these points, compute fx and fy, set each equal to zero, and solve the resulting system of algebraic equations. There can be one, several, or no stationary points, depending on the function.
求驻点时，先计算 fx 和 fy，令它们分别等于零，再解所得的代数方程组。根据函数的不同，可能有一个、多个驻点，也可能没有。
For example, f(x, y) = x³ – 3x + y² gives fx = 3x² – 3 = 0 ⇒ x = ±1, and fy = 2y = 0 ⇒ y = 0. Thus the stationary points are (1, 0) and (-1, 0).
例如，f(x, y) = x³ – 3x + y² 给出 fx = 3x² – 3 = 0 ⇒ x = ±1，以及 fy = 2y = 0 ⇒ y = 0。因此驻点为 (1, 0) 和 (-1, 0)。
8. Classifying Stationary Points Using the Hessian | 利用海塞矩阵对驻点进行分类
To determine the nature of a stationary point, we evaluate the second-order partial derivatives at that point and form the Hessian determinant D = fxx·fyy – (fxy)². The sign of D and the sign of fxx classify the point.
为确定驻点的性质，我们计算该点处的二阶偏导数值，并构成海塞判别式 D = fxx·fyy – (fxy)²。D 的符号和 fxx 的符号决定了点的类型。
If D > 0 and fxx > 0, the point is a local minimum. If D > 0 and fxx < 0, it is a local maximum. If D < 0, the point is a saddle point (a point where the surface curves up in one direction and down in another). If D = 0, the test is inconclusive and further investigation is needed.
若 D > 0 且 fxx > 0，则为局部极小值点。若 D > 0 且 fxx < 0，则为局部极大值点。若 D < 0，则为鞍点（曲面在一个方向上弯曲向上，另一方向弯曲向下的点）。若 D = 0，则判别法失效，需进一步分析。
In the previous example, at (1, 0): fxx = 6x = 6 > 0, fyy = 2, fxy = 0, so D = 6×2 – 0 = 12 > 0 ⇒ local minimum. At (-1, 0): fxx = -6 < 0, D = -6×2 = -12 < 0 ⇒ saddle point.
在前例中，对于 (1, 0)：fxx = 6x = 6 > 0，fyy = 2，fxy = 0，因此 D = 6×2 – 0 = 12 > 0 ⇒ 局部极小。对于 (-1, 0)：fxx = -6 < 0，D = -6×2 = -12 < 0 ⇒ 鞍点。
9. Applied Maxima and Minima Problems | 极大值与极小值的应用问题
Many real-world optimisation problems involve expressing a quantity, such as volume, cost, or surface area, as a function of two independent variables and then using partial derivatives to find its maximum or minimum value subject to no constraints or a simple constraint that can be eliminated.
许多现实中的优化问题都是先将某个量（如体积、成本、表面积）表示为两个自变量的函数，然后利用偏导数求出其无约束条件下的最大值或最小值；若存在简单约束，则将其消去。
Typical steps: model the problem as a function z = f(x, y); find stationary points by setting fx = 0, fy = 0; classify using the second derivative test; and interpret the result in the original context, checking boundary conditions if necessary.
典型步骤为：将问题建模为函数 z = f(x, y)；通过 fx = 0, fy = 0 求驻点；用二阶导数检验进行分类；在原始语境中解释结果，必要时检查边界条件。
For instance, a WJEC question might ask for the dimensions of a rectangular box with a fixed sum of length, width and height that maximises the volume. By eliminating one variable using the constraint, you reduce the problem to a function of two variables and then apply the techniques above.
例如，一道 WJEC 题目可能要求找出长、宽、高之和固定的长方体盒子的尺寸，使体积最大。利用约束条件消去一个变量后，问题转化为二元函数，然后应用上述方法。
Always check that the solution makes practical sense and explicitly state that you have confirmed it is a maximum (by D > 0 and fxx < 0 or by considering the nature of the function).
务必检查解在实际中是否合理，并明确说明已确认其为最大值（通过 D > 0 且 fxx < 0 或通过考量函数特性）。
10. Common Mistakes and Exam Tips | 常见错误与应试技巧
One frequent error is forgetting to treat one variable as a constant when finding a partial derivative. A good habit is to physically circle the variable you are holding constant before differentiating.
一个常见错误是求偏导数时忘记将另一个变量视为常数。一个好习惯是在求导前，把被视为常数的变量圈出来。
Another common slip occurs in the second derivative test: mixing up the formula for D or misinterpreting the sign of fxx. Use a memory aid such as “D positive and fxx positive gives a minimum” (think of a smiling curve shape).
另一个常见疏忽出现在二阶导数检验中：混淆 D 的计算公式，或误判 fxx 的符号。可使用记忆口诀，比如“D正且fxx正得极小”（想象一个开口向上的抛物线形状）。
When applying the chain rule, clearly label which variables are intermediate and which are independent. Writing out a tree diagram can help visualise the dependence structure and avoid missing terms.
应用链式法则时，要清楚标注哪些是中间变量，哪些是自变量。画出树形图有助于直观展示依赖结构，避免遗漏项。
During the exam, show all steps of solving fx = 0 and fy = 0, and do not attempt to guess the nature of stationary points without computing D. Always substitute values carefully and double-check signs.
考试中要展示解 fx = 0 与 fy = 0 的全部步骤，不要未经计算 D 就猜测驻点性质。代入数值时务必细心，反复核对符号。
Finally, practice a wide variety of functions – polynomials, trigonometric, exponential and logarithmic – as WJEC papers are known for mixing function types in a single question to test your adaptability.
最后，要练习多种类型的函数——多项式、三角、指数、对数——因为 WJEC 试卷常在一道题中混合不同函数类型，考察综合应变能力。
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