📚 A-Level Edexcel Maths: Partial Differentiation Essentials | A-Level Edexcel 数学：偏微分 考点精讲
Partial differentiation is a cornerstone of multivariable calculus and a compulsory topic in Edexcel A-Level Further Mathematics, specifically within the Further Pure 2 (FP2) module. It extends differentiation to functions of two or more variables, equipping you to analyse surfaces, rates of change, and constrained optimisation. This article breaks down every critical sub-topic, from first-order partial derivatives through Lagrange multipliers, with paired explanations in English and Chinese, fully aligned with the Edexcel syllabus.
偏微分是多变量微积分的基石，也是 Edexcel A-Level 进阶数学（特别是 Further Pure 2 模块）的必考内容。它将求导推广到含有两个或更多变量的函数，帮助你分析曲面、变化率和约束优化问题。本文拆解了从一阶偏导数到拉格朗日乘数法的所有关键子课题，并配以中英对照讲解，完全紧扣 Edexcel 考纲。
1. Introduction to Partial Differentiation | 偏微分简介
A partial derivative of a function f(x, y) measures how f changes as one variable varies while the others are held constant. The notation ∂f/∂x reads ‘partial derivative of f with respect to x’. Unlike ordinary derivatives, we use the curly symbol ∂ to emphasise that other variables are temporarily treated as constants.
对于二元函数 f(x, y)，偏导数 ∂f/∂x 描述的是在固定 y 的情况下，f 随 x 变化的变化率。符号 ∂ 读作“partial”，用以区别于一元函数的全导数。求偏导时，只需将其他自变量视作常数，运用一元微分法则即可。
2. First-Order Partial Derivatives | 一阶偏导数
To compute ∂f/∂x, treat y as a constant and differentiate with respect to x. Similarly, for ∂f/∂y, treat x as a constant. Always keep the standard differentiation rules at hand: power rule, product rule, chain rule, and derivatives of standard functions. An example makes this clear.
计算 ∂f/∂x 时，将 y 视为常数并对 x 求导；计算 ∂f/∂y 时则将 x 视为常数。牢记幂函数、乘积、链式法则以及基本初等函数的导数公式。下面是一个示例。
If f(x,y) = 3x²y³ + eˣ sin y, then ∂f/∂x = 6x y³ + eˣ sin y, and ∂f/∂y = 9x²y² + eˣ cos y.
若 f(x,y) = 3x²y³ + eˣ sin y，则 ∂f/∂x = 6x y³ + eˣ sin y，∂f/∂y = 9x²y² + eˣ cos y。
Notice how the term 3x²y³ becomes 6x y³ when differentiating with respect to x; y³ is kept untouched. When differentiating with respect to y, x² becomes a constant multiplier. This consistent treatment is the essence of partial differentiation.
注意对 x 求导时，y³ 被当作常量保留，3x² 求导得 6x；对 y 求导时，x² 成了常系数。这种“视其他变量为常数”的操作就是偏微分的核心。
3. Second-Order & Mixed Partial Derivatives | 二阶与混合偏导数
Once you have first-order partials, you can differentiate again to obtain second-order derivatives: ∂²f/∂x², ∂²f/∂y², and the mixed derivatives ∂²f/∂x∂y and ∂²f/∂y∂x. For sufficiently smooth functions (all standard A-Level functions satisfy this), Clairaut’s theorem guarantees that the mixed partials are equal: ∂²f/∂x∂y = ∂²f/∂y∂x.
计算出 ∂f/∂x 和 ∂f/∂y 后，可以继续求二阶偏导数：∂²f/∂x², ∂²f/∂y²，以及混合偏导数 ∂²f/∂x∂y 和 ∂²f/∂y∂x。对于满足光滑条件的函数（A-Level 涉及的所有函数均满足），混合偏导数相等——这就是克莱罗定理：∂²f/∂x∂y = ∂²f/∂y∂x。
Example: f(x,y) = x³y². ∂f/∂x = 3x²y², ∂²f/∂x² = 6x y². ∂f/∂y = 2x³y, ∂²f/∂y² = 2x³. ∂²f/∂x∂y = 6x²y, and ∂²f/∂y∂x = 6x²y.
示例：f(x,y) = x³y²。∂f/∂x = 3x²y², ∂²f/∂x² = 6x y²。∂f/∂y = 2x³y, ∂²f/∂y² = 2x³。∂²f/∂x∂y = 6x²y，且 ∂²f/∂y∂x = 6x²y。
This symmetry simplifies calculations and is a common check in exam problems.
这种对称性可简化计算，也是考试中常用的验算手段。
4. The Chain Rule | 链式法则
When f is a function of x and y, and both x and y depend on a single variable t (or on other intermediate variables), the total derivative with respect to t is given by the multivariable chain rule:
当函数 f 取决于 x 和 y，而 x 与 y 又同时依赖于某个变量 t（或其他中间变量）时，f 对 t 的全导数由多变量链式法则给出：
df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
If x and y themselves depend on two variables u and v, the partial derivatives of f with respect to u and v are constructed similarly. Always draw a dependency tree to avoid mistakes: f connected to x and y, then x, y connected to their variables.
建议画出变量依赖关系图：f → x, y；再连接 x, y → t（或 u, v）。切勿遗漏分支。常见的考试陷阱是求 ∂f/∂u 时遗漏 ∂f/∂y · ∂y/∂u 这一项。
5. Implicit Partial Differentiation | 隐函数的偏微分
When an equation F(x, y) = 0 defines y implicitly as a function of x, we can find dy/dx without solving for y explicitly:
当方程 F(x, y) = 0 隐含地定义了 y 是 x 的函数时，无需解出 y 的显式，即可求导：
dy/dx = – (∂F/∂x) / (∂F/∂y)
For instance, the circle x² + y² = 25 gives F(x,y)= x²+y²-25=0. Then ∂F/∂x = 2x, ∂F/∂y = 2y, so dy/dx = -x/y. This easily extends to F(x,y,z)=0, allowing us to find partial derivatives like (∂z/∂x) = – (∂F/∂x)/(∂F/∂z).
例如圆 x² + y² = 25，设 F(x,y)=x²+y²-25=0，则∂F/∂x=2x, ∂F/∂y=2y，故 dy/dx = -x/y。此法可推广至三元隐函数 F(x,y,z)=0，求得 ∂z/∂x = – (∂F/∂x)/(∂F/∂z)。
6. Total Differentials & Small Changes | 全微分与小变化
The total differential df approximates the change in f when x and y change by small amounts Δx and Δy. For f(x,y), the formula is:
全微分 df 给出了当自变量 x, y 分别发生微小变化 Δx, Δy 时，函数 f 的近似改变量。定义式为：
df = (∂f/∂x) dx + (∂f/∂y) dy, and Δf ≈ (∂f/∂x) Δx + (∂f/∂y) Δy.
This is extremely useful for error estimation. If a measurement x has a possible error of ±Δx, the propagated error in f is roughly |∂f/∂x · Δx| + |∂f/∂y · Δy| (taking absolute values for worst-case scenarios).
这在误差分析中极为有用。例如，若测量值 x 的最大绝对误差为 Δx，则 f 的传播最大误差约为 |∂f/∂x · Δx| + |∂f/∂y · Δy|（取绝对值得到最坏情况）。
7. Rates of Change | 变化率问题
Combine partial differentiation with the chain rule to relate the rates at which different quantities change with respect to time t. For example, the volume V of a cone with radius r and height h: V = ⅓πr²h. If r and h both change with time, the rate dV/dt is given by:
利用偏导数与链式法则，可以将多个物理量随时间 t 的变化率联系起来。比如圆锥体积 V = ⅓πr²h，如果底半径 r 和高 h 均随时间变化，则：
dV/dt = (∂V/∂r) (dr/dt) + (∂V/∂h) (dh/dt) = (⅔πrh) (dr/dt) + (⅓πr²) (dh/dt).
This technique appears regularly in Edexcel FP2 exam questions, often asking you to evaluate a specific rate when certain dimensions and rates are known.
这类问题在 Edexcel 考试中频繁出现，常常要求你在给定某些尺寸及其变化率时，计算另一量的变化速率。
8. Stationary Points of Two-Variable Functions | 二元函数的驻点
To find stationary points of f(x,y), set both first-order partial derivatives to zero simultaneously:
欲求二元函数 f(x,y) 的驻点，需令两个一阶偏导数同时为零：
∂f/∂x = 0 and ∂f/∂y = 0
Solve the resulting system of equations. Each solution (x₀, y₀) is a stationary point. It might be a local maximum, local minimum, or a saddle point.
解这个方程组，每一组解 (x₀, y₀) 即为一个驻点。它可能是局部极大点、局部极小点或鞍点。务必同时满足两个方程——漏掉一个条件是常见错误。
9. Classifying Stationary Points | 驻点分类（二阶导数检验）
Once a stationary point is found, compute the second derivatives and form the discriminant D:
找到驻点后，计算二阶偏导数并构造判别式 D：
D = (∂²f/∂x²)(∂²f/∂y²) – (∂²f/∂x∂y)²
Then at the stationary point: if D > 0 and ∂²f/∂x² > 0, it’s a local minimum; if D > 0 and ∂²f/∂x² < 0, a local maximum; if D < 0, a saddle point. When D = 0 the test is inconclusive. Memorise this condition carefully — it is tested almost every year.
若 D > 0 且 ∂²f/∂x² > 0，则为局部极小点；若 D > 0 且 ∂²f/∂x² < 0，则为局部极大点；若 D < 0，则为鞍点。当 D = 0 时无法判定。该判别法每年必考，务必熟记。
10. Lagrange Multipliers | 拉格朗日乘数法
For optimising f(x,y) subject to a constraint g(x,y)=0, set up the Lagrangian L(x,y,λ) = f(x,y) – λ g(x,y) (some texts use +λ, but the sign is irrelevant if you are solving for stationary points). Then solve:
在约束条件 g(x,y)=0 下求 f(x,y) 的极值，可构造拉格朗日函数 L(x,y,λ) = f(x,y) – λ g(x,y)（符号 + 亦可，最终解等价）。然后令偏导为零：
∂L/∂x = 0, ∂L/∂y = 0, ∂L/∂λ = 0 (which recovers g=0).
Solve these three equations simultaneously. The λ values themselves are rarely required; focus on the (x,y) coordinates. This method elegantly handles constraints like x²+y²=1, x+2y=5, etc., appearing frequently in maximum/minimum problems in FP2.
联立这三个方程求解（λ 的值通常不重要，关键是求得 x, y）。此法能优雅地处理诸如 x²+y²=1, x+2y=5 等约束下的极值问题，在 FP2 压轴题中常见。
11. Applications: Errors & Approximations | 应用：误差与近似
When measured quantities have uncertainties, the total differential gives a linear approximation of the maximum absolute or percentage error in a computed quantity. For Q = f(x,y), with errors Δx, Δy, the approximate error ΔQ is:
当测量量存在不确定度时，全微分可给出计算量 Q = f(x,y) 的最大绝对误差或相对误差的线性近似：
ΔQ ≈ (∂Q/∂x) Δx + (∂Q/∂y) Δy, and for percentage error: (ΔQ/Q) × 100%.
A typical exam question provides measured values and their possible errors, then asks for the maximum error in the final result. Always use absolute values of each term for the worst-case scenario unless asked otherwise.
典型考题会给出测量值及其误差，让你推算最终结果的最大误差。除非题目另有要求，通常将每项取绝对值以计算最坏情况。
12. Exam Tips & Common Pitfalls | 考试技巧与常见错误
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When finding ∂f/∂x, consciously circle or underline the variables you treat as constant. Many marks are lost by forgetting that y is constant and misapplying product rule.
求 ∂f/∂x 时，明确圈出视为常数的变量——经常有学生忘记 y 是常数，错误使用乘积法则。
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Mixed partial derivatives are equal only if the function is smooth. In Edexcel problems this always holds, but beware of any piecewise functions in future study.
在 Edexcel 考纲内，混合偏导数总是相等，但需留意分段函数可能存在反例。
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In the chain rule, always include all paths from the outer function to the differentiation variable. Draw a tree diagram to check you haven’t missed a branch.
应用链式法则时，务必画出依赖树，确保没有遗漏任一分支。
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For stationary points, solving ∂f/∂x=0 and ∂f/∂y=0 may yield several points. Substitute each into D carefully; a sign error in D can flip the classification.
求解驻点时，方程组可能有多组解，需逐一代入 D 检验；D 的符号算错一个便会判错极值类型。
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With Lagrange multipliers, don’t forget to include the constraint equation itself (∂L/∂λ=0). Many candidates solve the first two and miss the constraint, obtaining incorrect points.
使用拉格朗日乘数法时，别忘了第三个方程 ∂L/∂λ=0 本身——很多考生只解前两个
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