📚 A-Level Mathematics: Partial Differentiation Key Points | A-Level 数学：偏微分 考点精讲
Partial differentiation generalises ordinary differentiation to functions of several variables. In A-Level Mathematics (typically within Further Pure modules), you will compute partial derivatives, use them in optimisation and error analysis, and apply the chain rule in multivariable contexts. This article distils all core exam-ready knowledge.
偏微分将一元函数的求导推广到多元函数。在 A-Level 数学（通常出现在进阶纯数中）里，你需要计算偏导数，将其用于最优化和误差分析，并熟练运用链式法则。本文将提炼全部核心考点。
1. What is Partial Differentiation? | 偏微分是什么？
In ordinary calculus, for y = f(x), the derivative dy/dx measures the instantaneous rate of change of y with respect to x. When a function depends on two or more independent variables, say z = f(x, y), we define partial derivatives that measure the rate of change with respect to one variable while the others are held constant.
在普通微积分中，对于 y = f(x)，导数 dy/dx 度量了 y 随 x 变化的瞬时变化率。当一个函数依赖于两个或更多自变量时，例如 z = f(x, y)，我们定义偏导数来度量仅一个变量变化而其余变量保持不变时的变化率。
For instance, if z = x² + y², treating y as a constant gives ∂z/∂x = 2x; treating x as a constant gives ∂z/∂y = 2y. The curly ∂ symbol distinguishes a partial derivative from an ordinary derivative.
例如，若 z = x² + y²，将 y 视为常数得到 ∂z/∂x = 2x；将 x 视为常数得到 ∂z/∂y = 2y。弯体的 ∂ 符号用于区分偏导数和常导数。
A partial derivative answers the question: ‘How does the function change if we nudge only x, while freezing y?’ Geometrically, it gives the slope of the surface in the direction of the chosen axis.
偏导数回答的问题是：“若我们仅轻微改变 x 而冻结 y，函数如何变化？” 从几何上看，它给出了曲面在所选坐标轴方向的坡度。
2. First-Order Partial Derivatives | 一阶偏导数
To compute ∂z/∂x for z = f(x, y), differentiate with respect to x and treat all y-terms as constants. For z = x³ y + eˣ sin y:
∂z/∂x = 3x² y + eˣ sin y
计算 ∂z/∂x 时，对 x 求导并将所有含 y 的项视为常数。对于 z = x³ y + eˣ sin y：
∂z/∂x = 3x² y + eˣ sin y
To compute ∂z/∂y, treat x as constant:
∂z/∂y = x³ + eˣ cos y
计算 ∂z/∂y 时，将 x 视为常数：
∂z/∂y = x³ + eˣ cos y
Products or compositions that involve both variables are handled using the same rules (product rule, chain rule) while holding the other variable fixed. For z = x²eʸ + ln x · sin y:
∂z/∂x = 2x eʸ + (1/x) sin y
∂z/∂y = x² eʸ + ln x · cos y
涉及两个变量的乘积或复合运算时，同样使用求导法则（乘积法则、链式法则），同时固定另一个变量。对于 z = x²eʸ + ln x · sin y：
∂z/∂x = 2x eʸ + (1/x) sin y ; ∂z/∂y = x² eʸ + ln x · cos y
3. Notation and Interpretation | 记号与解释
Common notations for the first-order partial derivative of z = f(x, y) with respect to x include:
∂z/∂x, f_x(x, y), ∂f/∂x, and sometimes (∂z/∂x)_y to emphasise that y is held constant.
对 z = f(x, y) 关于 x 的一阶偏导数，常见记号包括：∂z/∂x、f_x(x, y)、∂f/∂x，有时也用 (∂z/∂x)_y 强调 y 被固定。
Geometrically, at a point (a, b), ∂z/∂x gives the gradient of the tangent line to the surface z = f(x, y) in the plane y = b. Similarly, ∂z/∂y gives the gradient in the plane x = a. These slopes help sketch the local behaviour of the surface.
几何上，在点 (a, b) 处，∂z/∂x 给出了曲面 z = f(x, y) 在平面 y = b 内的切线斜率。类似地，∂z/∂y 给出了平面 x = a 内的切线斜率。这些斜率有助于描绘曲面的局部形态。
Partial derivatives can also be denoted by subscripts, for example ∂²z/∂x² = f_xx and ∂²z/∂x∂y = f_xy. This compact form is useful when writing higher-order derivatives.
偏导数还可以用下标表示，例如 ∂²z/∂x² = f_xx，∂²z/∂x∂y = f_xy。在书写高阶导数时，这种简洁形式非常方便。
4. Second-Order Partial Derivatives | 二阶偏导数
Once you have first-order partial derivatives, you can differentiate them again to obtain second-order partial derivatives. For z = f(x, y), there are four second-order derivatives:
∂²z/∂x², ∂²z/∂y², ∂²z/∂x∂y, ∂²z/∂y∂x
得到一阶偏导数后，可以再次求导获得二阶偏导数。对于 z = f(x, y)，共有四个二阶偏导数：
∂²z/∂x²、∂²z/∂y²、∂²z/∂x∂y、∂²z/∂y∂x
For example, if z = x³y + 2xy², first compute ∂z/∂x = 3x²y + 2y² and ∂z/∂y = x³ + 4xy. Then:
∂²z/∂x² = 6xy,
∂²z/∂y² = 4x,
∂²z/∂x∂y = ∂/∂y (3x²y + 2y²) = 3x² + 4y,
∂²z/∂y∂x = ∂/∂x (x³ + 4xy) = 3x² + 4y
例如，若 z = x³y + 2xy²，先求 ∂z/∂x = 3x²y + 2y² 和 ∂z/∂y = x³ + 4xy。然后求二阶偏导：
∂²z/∂x² = 6xy、∂²z/∂y² = 4x、∂²z/∂x∂y = 3x² + 4y、∂²z/∂y∂x = 3x² + 4y
5. Mixed Partial Derivatives and Clairaut’s Theorem | 混合偏导数与克莱罗定理
The two mixed partial derivatives ∂²z/∂x∂y and ∂²z/∂y∂x are not automatically equal, but Clairaut’s Theorem states that if they are continuous on a region, then they are equal:
∂²z/∂x∂y = ∂²z/∂y∂x
两个混合偏导数 ∂²z/∂x∂y 和 ∂²z/∂y∂x 并非自动相等，但克莱罗定理指出：若它们在区域内连续，则两者相等：
∂²z/∂x∂y = ∂²z/∂y∂x
In A-Level exams you can safely assume equality for all standard functions (polynomials, exponentials, trigonometric functions, logarithms, etc.). The theorem simplifies the calculation of second-order derivatives and is essential when applying the second derivative test for stationary points.
在 A-Level 考试中，对于所有标准函数（多项式、指数、三角函数、对数等），你都可以放心地假定混合偏导相等。该定理简化了二阶导数的计算，并在驻点分类的二阶导数检验中至关重要。
Use this property to check your work: if you compute f_xy and f_yx and get different results, you have likely made an algebraic mistake.
利用这一性质可以核查你的计算：若你算出 f_xy 和 f_yx 不同，很可能出现了代数错误。
6. The Total Differential | 全微分
The total differential dz approximates the change in z = f(x, y) when both x and y change by small amounts dx and dy:
dz = (∂z/∂x) dx + (∂z/∂y) 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) 的整体变化量：
dz = (∂z/∂x) dx + (∂z/∂y) dy
For example, if the volume of a cylinder V = πr²h, then dV = 2πrh dr + πr² dh. This approximates the error in volume when the radius and height have measurement errors dr and dh.
例如，圆柱体体积 V = πr²h 的全微分为 dV = 2πrh dr + πr² dh。当半径与高度的测量误差分别为 dr 和 dh 时，该式可估计体积的误差。
In error analysis, you often replace dr and dh with absolute errors and take absolute values to bound the maximum possible error: |ΔV| ≈ |∂V/∂r| |Δr| + |∂V/∂h| |Δh|.
在误差分析中，通常将 dr 和 dh 替换为绝对误差并取绝对值来估算最大可能误差：|ΔV| ≈ |∂V/∂r| |Δr| + |∂V/∂h| |Δh|。
7. The Chain Rule for Partial Derivatives | 偏导数的链式法则
If z = f(x, y) and both x and y depend on a single variable t, i.e. x = x(t), y = y(t), then the ordinary derivative dz/dt is given by:
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
若 z = f(x, y) 且 x 和 y 都依赖于单个变量 t（即 x = x(t)、y = y(t)），则常导数 dz/dt 由下式给出：
dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
If x and y themselves depend on two variables s and t, say x = x(s, t) and y = y(s, t), then:
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t)
如果 x 和 y 又依赖于两个变量 s 和 t，例如 x = x(s, t)、y = y(s, t)，则：
∂z/∂s = (∂z/∂x)(∂x/∂s) + (∂z/∂y)(∂y/∂s)
∂z/∂t = (∂z/∂x)(∂x/∂t) + (∂z/∂y)(∂y/∂t)
A useful mnemonic is to draw a dependency tree: z connected to x and y, and then x and y connected to the new variables. The derivative along any branch is the product of the derivatives along the path, and all paths are summed.
一个有用的记忆法是画出依赖关系树：z 连接到 x 和 y，然后 x 和 y 再连接到新的变量。任一路径上的导数就是该路径各段导数的乘积，再把所有路径求和。
8. Implicit Partial Differentiation | 隐函数偏微分
When an equation F(x, y, z) = 0 implicitly defines z as a function of x and y, you can find the partial derivatives of z without solving explicitly. Assuming ∂F/∂z ≠ 0:
∂z/∂x = – (∂F/∂x) / (∂F/∂z)
∂z/∂y = – (∂F/∂y) / (∂F/∂z)
当方程 F(x, y, z) =
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