📚 Partial Derivatives | 偏导数
Partial derivatives are a fundamental concept in multivariable calculus, building upon the derivative rules learned in single-variable calculus. For IB Mathematics: Analysis and Approaches (HL) students, partial derivatives are essential for analysing functions of several variables, finding optimum values under constraints, and solving a wide range of applied problems. This article provides a thorough revision of partial derivatives, covering definitions, key rules, geometric intuition, and practical applications.
偏导数是多元微积分中的一个基本概念，建立在单变量微积分导数的规则之上。对于IB数学：分析与方法（HL）学生来说，偏导数对于分析多元函数、在约束下寻找最优值以及解决各种应用问题至关重要。本文将全面复习偏导数，涵盖定义、关键规则、几何直观以及实际应用。
1. Definition of Partial Derivatives | 偏导数的定义
For a function of two variables, z = f(x, y), the partial derivative with respect to x is defined as the limit ∂f/∂x = limh→0 [f(x+h, y) – f(x, y)] / h, provided the limit exists. Geometrically, this is the rate of change of f as x varies while y is held constant.
对于二元函数 z = f(x, y)，对 x 的偏导数定义为极限 ∂f/∂x = limh→0 [f(x+h, y) – f(x, y)] / h，前提是该极限存在。从几何上看，这是当 y 保持不变、x 变化时 f 的变化率。
Similarly, the partial derivative with respect to y is ∂f/∂y = limh→0 [f(x, y+h) – f(x, y)] / h, treating x as a constant. If both partial derivatives exist at a point, the function is said to be differentiable there, implying continuity.
类似地，对 y 的偏导数为 ∂f/∂y = limh→0 [f(x, y+h) – f(x, y)] / h，将 x 视为常数。如果某点这两个偏导数都存在，则函数在该点可微，从而蕴含连续性。
∂f/∂x = fx, ∂f/∂y = fy
2. Notation and Computation | 记号和计算
Common notations for partial derivatives include ∂z/∂x, ∂f/∂x, fx, and zx. When computing a partial derivative, all other independent variables are treated as constants, and standard differentiation rules (power rule, product rule, chain rule) are applied.
偏导数的常用记号包括 ∂z/∂x、∂f/∂x、fx 和 zx。计算偏导数时，将所有其他自变量视为常数，并应用标准的微分法则（幂法则、乘积法则、链式法则）。
Example: For f(x, y) = x²y + 3xy³, ∂f/∂x = 2xy + 3y³ (treat y constant), and ∂f/∂y = x² + 9xy² (treat x constant). Always differentiate term by term.
例：对于 f(x, y) = x²y + 3xy³，∂f/∂x = 2xy + 3y³（将 y 当作常数），而 ∂f/∂y = x² + 9xy²（将 x 当作常数）。需要逐项求导。
For more complex expressions, such as f(x, y) = e^(xy) sin(x+y), partial derivatives require careful application of product and chain rules while holding the other variable constant.
对于更复杂的表达式，如 f(x, y) = e^(xy) sin(x+y)，偏导数的计算需要在另一个变量保持不变的情况下，仔细运用乘积法则和链式法则。
3. Geometric Interpretation | 几何意义
The graph of z = f(x, y) is a surface in three-dimensional space. Fixing y = b gives a curve z = f(x, b) on this surface, and the partial derivative ∂f/∂x at (a, b) is the slope of the tangent line to that curve in the x-direction. Similarly, ∂f/∂y gives the slope in the y-direction.
z = f(x, y) 的图像是三维空间中的一个曲面。固定 y = b 得到该曲面上的曲线 z = f(x, b)，偏导数 ∂f/∂x 在 (a, b) 处的值就是该曲线沿 x 方向的切线斜率。类似地，∂f/∂y 给出沿 y 方向的斜率。
Thus, the two partial derivatives represent the rates of change along two orthogonal directions parallel to the coordinate axes. The tangent plane to the surface at a point can be expressed using these slopes.
因此，这两个偏导数代表了沿坐标轴平行方向的正交方向上的变化率。曲面在某点的切平面可以用这些斜率表示出来。
4. Higher-Order Partial Derivatives | 高阶偏导数
Just as we take second derivatives of single-variable functions, we can compute second-order partial derivatives: ∂²f/∂x² (differentiate twice with respect to x), ∂²f/∂y², and mixed partials ∂²f/∂x∂y and ∂²f/∂y∂x.
正如我们对单变量函数求二阶导数一样，我们也可以计算二阶偏导数：∂²f/∂x²（对 x 偏导两次）、∂²f/∂y² 以及混合偏导数 ∂²f/∂x∂y 和 ∂²f/∂y∂x。
∂²f/∂x∂y = ∂/∂x (∂f/∂y), ∂²f/∂y∂x = ∂/∂y (∂f/∂x)
Clairaut’s theorem states that if the second-order mixed partials are continuous on a region, then ∂²f/∂x∂y = ∂²f/∂y∂x. This symmetry is very useful in verification of computations and in thermodynamics.
克莱罗定理指出，若二阶混合偏导数在区域上连续，则 ∂²f/∂x∂y = ∂²f/∂y∂x。这种对称性在验算以及在热力学中非常有用。
Higher-order partials appear in Taylor expansions for functions of two variables and in the classification of stationary points.
更高阶的偏导数出现在二元函数的泰勒展开式以及驻点的分类中。
5. The Chain Rule for Partial Derivatives | 偏导数的链式法则
Suppose u = f(x, y) and both x and y are functions of another variable t: x = g(t), y = h(t). Then the total derivative of u with respect to t is given by du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt). This is the multivariable chain rule.
假设 u = f(x, y)，而 x 和 y 都是另一变量 t 的函数：x = g(t)，y = h(t)。那么 u 对 t 的全导数为 du/dt = (∂u/∂x)(dx/dt) + (∂u/∂y)(dy/dt)。这就是多元链式法则。
If x and y themselves depend on two parameters s and t, then ∂u/∂s = (∂u/∂x)(∂x/∂s) + (∂u/∂y)(∂y/∂s), and similarly for ∂u/∂t. This tree-diagram approach is essential for correctly computing derivatives in multivariable contexts.
若 x 和 y 本身依赖于两个参数 s 和 t，则 ∂u/∂s = (∂u/∂x)(∂x/∂s) + (∂u/∂y)(∂y/∂s)，∂u/∂t 类似。这种树状图方法对于在多元情况下正确计算导数至关重要。
6. Implicit Differentiation | 隐函数微分法
When an equation F(x, y, z) = 0 defines z implicitly as a function of x and y, we can find partial derivatives without solving for z explicitly. Using the chain rule, assuming F differentiable and ∂F/∂z ≠ 0, we get ∂z/∂x = – (∂F/∂x) / (
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