📚 IB Math: Partial Derivatives – Key Concepts and Exam Tips | IB 数学：偏微分 考点精讲
In IB Mathematics, especially at the Higher Level, extending differentiation to functions of two or more variables is a powerful step beyond single-variable calculus. Partial derivatives allow us to explore how a multivariable function changes when only one of its inputs varies, keeping the others fixed. This topic underpins tangent plane approximation, optimization in space, and connects to advanced applications such as Lagrange multipliers. Mastering partial differentiation is essential not only for exam success but also for future studies in physics, engineering, and economics.
在 IB 数学中，尤其是在高水平课程，将微分推广到两个或多个变量的函数是超越单变量微积分的重要一步。偏微分使我们能够探究当只有一个自变量变化而其他变量保持不变时，多元函数如何变化。该主题支撑了切平面近似、空间中的最优化，并与拉格朗日乘数法等高级应用相联系。掌握偏微分不仅对考试成功至关重要，也为将来学习物理、工程和经济学打下基础。
1. Introduction to Multivariable Functions | 多元函数引言
A multivariable function accepts two or more independent inputs and produces a single output. In IB, we often work with surfaces represented by z = f(x, y), where the height z depends on horizontal coordinates x and y. Understanding the domain and range of such functions is the first step before discussing rates of change.
多元函数接受两个或多个自变量并产生一个输出。在 IB 中，我们经常处理由 z = f(x, y) 表示的曲面，其中高度 z 取决于水平坐标 x 和 y。在讨论变化率之前，理解此类函数的定义域和值域是第一步。
Visualising these functions through contour plots or 3D graphs helps build intuition. For example, a simple function like f(x, y) = x² + y² generates a paraboloid, and its contour curves are circles.
通过等高线图或三维图形对这些函数进行可视化有助于建立直觉。例如，一个简单的函数 f(x, y) = x² + y² 生成一个抛物面，其等高线为圆。
2. Understanding Partial Derivatives | 偏微分的定义
The partial derivative of f with respect to x, denoted ∂f/∂x, is the rate at which f changes as x varies while y is held constant. Formally, it is defined as the limit:
f 对 x 的偏微分，记作 ∂f/∂x，是在 y 保持不变时 f 随 x 变化的速率。正式定义为极限：
∂f/∂x = lim_{h → 0} (f(x + h, y) − f(x, y)) / h
Similarly, the partial derivative with respect to y, ∂f/∂y, is obtained by treating x as constant and differentiating with respect to y. The limit definition mirrors the standard derivative but restricts variation to one coordinate at a time.
类似地，对 y 的偏微分 ∂f/∂y，通过将 x 视为常数并对 y 求导得到。极限定义与普通导数一致，但每次只允许一个坐标变化。
3. Notation for Partial Derivatives | 偏微分的符号
Several notations appear in IB. Besides the Leibniz form ∂f/∂x, you may see fₓ or f_y. The subscript notation is compact: fₓ means ∂f/∂x and fₓₓ means ∂²f/∂x². When time or other parameters are involved, ∂/∂t is used.
IB 课程中会出现几种符号。除了莱布尼茨形式 ∂f/∂x，你可能看到 fₓ 或 f_y。下标符号更简洁：fₓ 表示 ∂f/∂x，fₓₓ 表示 ∂²f/∂x²。当涉及时间或其他参数时，会用 ∂/∂t。
The curly d (∂) distinguishes partial derivatives from ordinary derivatives (d). Always use ∂ when the function depends on more than one variable and we are differentiating with respect to one while holding others fixed.
曲线 d (∂) 将偏微分与常微分 (d) 区分开来。当函数依赖于多个变量且我们固定其他变量对一个变量求导时，一定要使用 ∂。
4. Computing Partial Derivatives | 计算偏微分
To find ∂f/∂x, treat every occurrence of y as a constant and differentiate with respect to x using standard rules (power, product, chain, etc.). For instance, if f(x, y) = x³y + sin(xy), then:
为求 ∂f/∂x，将出现的每个 y 都当作常数处理，并使用标准法则（幂法则、乘积法则、链式法则等）对 x 求导。例如，若 f(x, y) = x³y + sin(xy)，则：
∂f/∂x = 3x²y + y cos(xy)
For ∂f/∂y, treat x as constant: ∂f/∂y = x³ + x cos(xy). Common mistakes include forgetting to apply the chain rule to terms like sin(xy) when differentiating with respect to x (the derivative of xy with respect to x is y).
求 ∂f/∂y 时，将 x 当作常数：∂f/∂y = x³ + x cos(xy)。常见错误包括在求对 x 的导数时忘记对 sin(xy) 项应用链式法则（xy 对 x 的导数为 y）。
5. Geometric Interpretation | 几何意义
At a point (a, b) on a surface z = f(x, y), the partial derivative fₓ(a, b) is the slope of the tangent line to the curve formed by intersecting the surface with the plane y = b. That line has direction parallel to the x-axis. Thus fₓ gives the instantaneous rate of change of z as we move east/west.
在曲面 z = f(x, y) 上的一点 (a, b) 处，偏导数 fₓ(a, b) 是曲面与平面 y = b 相交所形成曲线的切线斜率。该切线与 x 轴平行。因此 fₓ 给出了当我们沿东西方向移动时 z 的瞬时变化率。
Similarly, f_y(a, b) is the slope of the tangent line to the curve lying in the plane x = a, measuring the rate of change in the north/south direction. Together, these slopes define the tangent plane.
类似地，f_y(a, b) 是位于平面 x = a 内的曲线的切线斜率，衡量南北方向的变化率。这两个斜率共同定义了切平面。
6. Higher-Order Partial Derivatives | 高阶偏微分
Since partial derivatives are themselves functions of x and y, we can differentiate them again. The second-order partial derivatives are:
因为偏导数本身也是 x 和 y 的函数，我们可以再次对它们求导。二阶偏导数为：
- ∂²f/∂x² = fₓₓ: differentiate ∂f/∂x with respect to x again — 再次对 x 求 ∂f/∂x 的导数
- ∂²f/∂y² = f_yy: differentiate ∂f/∂y with respect to y again — 再次对 y 求 ∂f/∂y 的导数
- ∂²f/∂x∂y = fₓ_y: first take ∂/∂x, then ∂/∂y of the result — 先求对 x 的偏导，再对结果求 y 的偏导
- ∂²f/∂y∂x = f_yₓ: first ∂/∂y, then ∂/∂x — 先求对 y 的偏导，再对结果求 x 的偏导
7. Mixed Partial Derivatives and Clairaut’s Theorem | 混合偏导与克莱罗定理
In most IB problems, the function is well-behaved (second-order partials are continuous). Clairaut’s Theorem states that the mixed partials are equal:
在大多数 IB 题目中，函数性质良好（二阶偏导数连续）。克莱罗定理指出混合偏导数相等：
∂²f/∂x∂y = ∂²f/∂y∂x
This equality allows us to compute the easier order without worrying about sequence. For f(x, y) = x²eʸ, we find ∂f/∂x = 2x eʸ, then ∂²f/∂y∂x = ∂/∂y (2x eʸ) = 2x eʸ. Doing ∂f/∂y = x² eʸ first, then ∂²f/∂x∂y = 2x eʸ gives the same result.
这个等式使我们能够计算较简单的顺序而无需担心次序。对于 f(x, y) = x²eʸ，我们求得 ∂f/∂x = 2x eʸ，然后 ∂²f/∂y∂x = ∂/∂y (2x eʸ) = 2x eʸ。若先求 ∂f/∂y = x² eʸ，再求 ∂²f/∂x∂y = 2x eʸ 得到相同结果。
8. The Chain Rule for Partial Derivatives | 偏微分的链式法则
When z = f(x, y) and both x and y are functions of another variable t, the total derivative dz/dt is given by:
当 z = f(x, y) 且 x 和 y 都是另一个变量 t 的函数时，全导数 dz/dt 由下式给出：
dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)
If x and y themselves depend on two variables u and v, we have a tree of dependencies. For instance, ∂z/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u). Drawing a dependency diagram is strongly recommended to apply the chain rule correctly in IB exam questions.
如果 x 和 y 本身依赖于两个变量 u 和 v，我们会有依赖树。例如，∂z/∂u = (∂f/∂x)(∂x/∂u) + (∂f/∂y)(∂y/∂u)。在 IB 考试题目中强烈建议画出依赖图以正确应用链式法则。
9. Tangent Planes and Linear Approximation | 切平面与线性近似
The equation of the tangent plane to the surface z = f(x, y) at the point (x₀, y₀, f(x₀, y₀)) is:
曲面 z = f(x, y) 在点 (x₀, y₀, f(x₀, y₀)) 处的切平面方程为：
z − f(x₀, y₀) = fₓ(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀)
This formula generalises the tangent line approximation from single-variable calculus. The tangent plane provides the best linear approximation to the function near the point, and it is often used to estimate small changes Δz ≈ fₓ Δx + f_y Δy.
这个公式是从单变量微积分中的切线近似推广而来的。切平面提供了函数在该点附近的最佳线性近似，常用来估计微小变化 Δz ≈ fₓ Δx + f_y Δy。
10. Optimization and Critical Points | 最优化与临界点
To locate local maxima, minima, or saddle points of a function f(x, y), we first solve for critical points where both first partials vanish: fₓ = 0 and f_y = 0 simultaneously.
为了找到函数 f(x, y) 的局部极大值、极小值或鞍点，我们首先求解两个一阶偏导数同时为零的点，即临界点：fₓ = 0 和 f_y = 0。
The nature of each critical point is then classified using the second derivative test. Compute D = fₓₓ f_yy − (fₓ_y)² at the critical point. If D > 0 and fₓₓ > 0, we have a local minimum; if D > 0 and fₓₓ < 0, a local maximum; if D < 0, the point is a saddle point. When D = 0, the test is inconclusive.
然后使用二阶导数判别法对每个临界点进行分类。在临界点计算 D = fₓₓ f_yy − (fₓ_y)²。如果 D > 0 且 fₓₓ > 0，则为局部极小值；如果 D > 0 且 fₓₓ < 0，则为局部极大值；如果 D < 0，则为鞍点；当 D = 0 时，判别法无效。
11. Lagrange Multipliers (brief) | 拉格朗日乘数法（简要）
IB Higher Level may introduce Lagrange multipliers to find extremum values of f(x, y) subject to a constraint g(x, y) = k. The method sets ∇f = λ ∇g, where ∇f = (fₓ, f_y) and ∇g = (gₓ, g_y). This yields the system:
IB 高水平课程可能引入拉格朗日乘数法来求 f(x, y) 在约束条件 g(x, y) = k 下的极值。该方法令 ∇f = λ ∇g，其中 ∇f = (fₓ, f_y)，∇g = (gₓ, g_y)。由此得到方程组：
fₓ = λ gₓ, f_y = λ g_y, g(x, y) = k
Solving these simultaneously finds the optimal points. This powerful technique appears in optimisation problems with limited resources.
联立求解这些方程可找出极值点。这种强大的技巧出现在资源有限的优化问题中。
12. Exam Tips and Common Mistakes | 考试技巧与常见错误
- Always indicate which variable is held constant — Write ∂f/∂x clearly, not df/dx, as mixing notations loses marks. — 始终标明哪个变量被固定——清晰地写出 ∂f/∂x 而不是 df/dx，混淆符号会失分。
- Check symmetry of mixed partials — Use Clairaut’s theorem to simplify calculations and verify your results. — 检查混合偏导的对称性——使用克莱罗定理简化计算并验证结果。
- Draw dependency diagrams for chain rule problems — Visualising the flow of variables reduces errors in setting up partial derivatives. — 为链式法则问题绘制依赖图——将变量流向可视化能减少建立偏导表达式时的错误。
- Don’t forget to test boundary points in optimization — In a closed domain, extrema can occur on the boundary, not just at interior critical points. — 不要忘记检验边界点——在有界区域上，极值可能出现在边界上，而不仅仅在内部临界点。
- Practice interpreting the second derivative test — Be comfortable calculating D and remembering the cases for min, max, and saddle. — 练习解释二阶导数判别法——熟练计算 D 并牢记极小、极大和鞍点的情况。
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