Partial Derivatives: Concepts and Computation | 偏导数的概念与计算

📚 Partial Derivatives: Concepts and Computation | 偏导数的概念与计算

In multivariable calculus, partial derivatives extend the idea of a derivative to functions of several variables. When you hold all but one variable constant and differentiate with respect to that single variable, you obtain a partial derivative. This concept is central to IB Mathematics Analysis and Approaches HL and is also essential in physics, economics, and engineering.

在多变量微积分中，偏导数将导数的思想推广到包含多个自变量的函数。当你固定除一个变量以外的所有变量，仅对这一个变量求导时，得到的就是偏导数。这个概念是IB数学分析与方法HL的核心内容，在物理、经济与工程中也至关重要。

1. Why Do We Need Partial Derivatives? | 为什么需要偏导数？

Consider a function of two variables, such as the temperature on a flat metal plate, written as T(x, y). At a given point (x₀, y₀), the temperature can change when you move in the x-direction, in the y-direction, or along any other direction in the plane. A single ordinary derivative cannot fully describe this variation. Partial derivatives give us a way to measure the rate of change in each coordinate direction independently.

考虑一个二元函数，例如一块金属板上的温度分布，记为 T(x, y)。在给定点 (x₀, y₀) 处，温度会随着沿x方向、y方向或平面内任何其他方向移动而改变。单独一个普通导数无法完整描述这种变化。偏导数使我们能够分别测量沿每个坐标方向的变化率。

For example, the partial derivative ∂f/∂x tells us how f changes when only x changes, while y is fixed. Similarly, ∂f/∂y tells us how f changes when only y changes, while x is fixed. This coordinate-wise approach is both simple and powerful.

例如，偏导数 ∂f/∂x 告诉我们当只有x变化而y保持不变时f如何变化；类似地，∂f/∂y 告诉我们当只有y变化而x保持不变时f如何变化。这种逐坐标处理的方式既简单又强大。

2. Formal Definition of a Partial Derivative | 偏导数的严格定义

Let f(x, y) be a real-valued function of two variables. The partial derivative of f with respect to x at the point (x₀, y₀) is defined as:

设 f(x, y) 是一个二元实值函数。函数 f 在点 (x₀, y₀) 处关于 x 的偏导数定义为：

∂f/∂x (x₀, y₀) = lim_{h→0} [f(x₀ + h, y₀) − f(x₀, y₀)] / h

provided this limit exists. Similarly, the partial derivative with respect to y is:

前提是这个极限存在。类似地，关于 y 的偏导数为：

∂f/∂y (x₀, y₀) = lim_{k→0} [f(x₀, y₀ + k) − f(x₀, y₀)] / k

Notice that in the first limit, y₀ is held fixed; in the second limit, x₀ is held fixed. The notation ∂f/∂x is read as “partial dee f by dee x”, and it is never written as df/dx for a multivariable function.

注意，在第一个极限中 y₀ 保持不变；在第二个极限中 x₀ 保持不变。记号 ∂f/∂x 读作”偏f比偏x”，对于多元函数不能写成 df/dx。

3. Geometric Interpretation | 几何意义

For a surface z = f(x, y), the partial derivative ∂f/∂x at a point gives the slope of the tangent line to the curve formed by intersecting the surface with the plane y = y₀. This tangent line lies in that vertical plane and measures how steeply the surface rises as you move in the positive x-direction.

对于曲面 z = f(x, y)，在一点处的偏导数 ∂f/∂x 表示曲面与平面 y = y₀ 相交所得曲线在该点的切线斜率。这条切线位于该竖直平面内，度量了当你沿x正方向移动时曲面上升的陡峭程度。

Similarly, ∂f/∂y gives the slope of the tangent line to the curve formed by intersecting the surface with the plane x = x₀. These two tangent lines together determine the tangent plane to the surface at that point, provided the partial derivatives are continuous.

类似地，∂f/∂y 表示曲面与平面 x = x₀ 相交所得曲线的切线斜率。当偏导数连续时，这两条切线共同确定了曲面在该点的切平面。

In IB problems, you may be asked to sketch these intersection curves or to interpret the sign of a partial derivative in a real-world context. A positive ∂f/∂x means f increases as x increases while y stays fixed.

在IB题目中，你可能会被要求画出这些交线，或在真实情境中解释偏导数的正负。∂f/∂x 为正意味着当x增大而y固定时，f增大。

4. Basic Computation Rules | 基本计算法则

To compute a partial derivative, treat every other variable as a constant and use all the ordinary differentiation rules: power rule, product rule, quotient rule, and chain rule.

计算偏导数时，把其他所有变量看作常数，并运用所有普通的求导法则：幂法则、乘积法则、商法则和链式法则。

Example 1: For f(x, y) = x³ y² + sin(xy), find ∂f/∂x and ∂f/∂y.

例1： 对于 f(x, y) = x³ y² + sin(xy)，求 ∂f/∂x 和 ∂f/∂y。

Holding y constant, we differentiate term by term:

固定 y 为常数，逐项求导：

∂f/∂x = 3x² y² + y cos(xy)

Here, d/dx [sin(xy)] = y cos(xy) because y is treated as a constant inside the argument xy.

这里，d/dx [sin(xy)] = y cos(xy)，因为 y 在 xy 中被视为常数。

Holding x constant gives:

固定 x 为常数，得到：

∂f/∂y = 2x³ y + x cos(xy)

Notice that the product x³ y² is differentiated with respect to y, so the x³ part remains untouched while the derivative of y² is 2y.

注意，对 x³ y² 关于 y 求导时，x³ 部分保持不变，而 y² 的导数为 2y。

5. Evaluating Partial Derivatives at a Point | 在一点处求偏导数值

Once you have an expression for a partial derivative, you can substitute the given point to find its numeric value. For example, for f(x, y) = e^{xy}, we have ∂f/∂x = y e^{xy}. At the point (2, 3), this becomes 3 e⁶.

一旦得到偏导数的表达式，就可以代入给定点求出数值。例如，对于 f(x, y) = e^{xy}，我们有 ∂f/∂x = y e^{xy}。在点 (2, 3) 处，该值变为 3 e⁶。

In some questions, you do not need to find the full partial derivative first. If only one point is required, you can substitute the fixed variable value into f before differentiating. For instance, to find ∂f/∂x at (2, 3), first set y = 3 in f(x, y) = x² + xy + y², giving f(x, 3) = x² + 3x + 9. Then differentiate with respect to x: f_x(x, 3) = 2x + 3, so f_x(2, 3) = 7.

有些题目不需要先求出完整的偏导数。如果只需求某一点的值，可以先代入选定变量的常数值，再对另一个变量求导。例如，要计算 ∂f/∂x 在 (2, 3) 处的值，先令 f(x, y) = x² + xy + y² 中的 y = 3，得到 f(x, 3) = x² + 3x + 9，再对x求导得 f_x(x, 3) = 2x + 3，所以 f_x(2, 3) = 7。

This method is often faster and reduces the chance of algebra errors in examination settings.

这种方法往往更快，并且在考试中减少代数运算出错的机会。

6. Higher-Order Partial Derivatives | 高阶偏导数

Just as with ordinary derivatives, you can take partial derivatives of partial derivatives. For a function f(x, y), the second-order partial derivatives are usually denoted as:

与普通导数类似，可以对偏导数再求偏导。对于函数 f(x, y)，二阶偏导数通常记为：

∂²f/∂x² = f_xx: differentiate f with respect to x twice.
∂²f/∂y² = f_yy: differentiate f with respect to y twice.
∂²f/∂x∂y = f_xy: first differentiate with respect to y, then with respect to x. (Be careful about the order!)
∂²f/∂y∂x = f_yx: first differentiate with respect to x, then with respect to y.

∂²f/∂x² = f_xx：对 f 关于 x 求导两次。
∂²f/∂y² = f_yy：对 f 关于 y 求导两次。
∂²f/∂x∂y = f_xy：先关于 y 求导，再关于 x 求导。（注意顺序！）
∂²f/∂y∂x = f_yx：先关于 x 求导，再关于 y 求导。

Example 2: Let f(x, y) = x³ y + y². Compute f_xy.

例2： 设 f(x, y) = x³ y + y²，计算 f_xy。

First differentiate with respect to y: f_y = x³ + 2y. Then differentiate with respect to x: f_xy = 3x².

先对 y 求导：f_y = x³ + 2y。再对 x 求导：f_xy = 3x²。

If instead we compute f_yx, we first differentiate with respect to x: f_x = 3x² y. Then differentiate with respect to y: f_yx = 3x². In this case they are equal.

如果改为计算 f_yx，先对x求导：f_x = 3x² y。再对y求导：f_yx = 3x²。本例中两者相等。

7. Equality of Mixed Partial Derivatives | 混合偏导数相等定理

For most functions encountered in IB, the mixed partial derivatives f_xy and f_yx are equal. This is guaranteed by Clairaut’s theorem, which states that if f_xy and f_yx are continuous near a point, then f_xy(a, b) = f_yx(a, b).

对于IB中遇到的大多数函数，混合偏导数 f_xy 与 f_yx 相等。这由克莱罗定理保证：如果 f_xy 和 f_yx 在一点附近连续，则 f_xy(a, b) = f_yx(a, b)。

This property is very useful: it allows you to choose the easier order of differentiation, and it can help you check your answers. It is also the basis for the symmetry of the Hessian matrix, which appears in optimisation problems.

这个性质非常有用：它允许你选择更简单的求导顺序，也可以帮助检验答案。它还是Hessian矩阵对称性的基础，而Hessian矩阵出现在最优化问题中。

However, the equality does not always hold for arbitrary functions, especially if the second derivatives are discontinuous. In IB, such irregular functions are rarely tested, but you should be aware of the theorem’s conditions.

然而，对于任意函数，尤其当二阶导数不连续时，该等式并不总成立。在IB中很少考到这种不规则函数，但你应该了解定理成立的条件。

8. Chain Rule for Partial Derivatives | 偏导数的链式法则

If z = f(x, y) and x = g(s, t), y = h(s, t), then the partial derivatives of z with respect to s and t are given by:

若 z = f(x, y)，且 x = g(s, t)，y = h(s, t)，则 z 关于 s 和 t 的偏导数由下式给出：

∂z/∂s = ∂z/∂x · ∂x/∂s + ∂z/∂y · ∂y/∂s

∂z/∂t = ∂z/∂x · ∂x/∂t + ∂z/∂y · ∂y/∂t

This is the multivariable version of the chain rule. It sums the contributions from each intermediate variable x and y.

这是多变量版本的链式法则。它将来自中间变量 x 和 y 的贡献加在一起。

Example 3: Let z = x² y, where x = s + t and y = s − t. Find ∂z/∂s.

例3： 设 z = x² y，其中 x = s + t，y = s − t。求 ∂z/∂s。

We have ∂z/∂x = 2xy, ∂z/∂y = x², ∂x/∂s = 1, ∂y/∂s = 1. Thus:

我们有 ∂z/∂x = 2xy，∂z/∂y = x²，∂x/∂s = 1，∂y/∂s = 1。因此：

∂z/∂s = 2xy · 1 + x² · 1 = 2xy + x²

Substituting x = s + t and y = s − t gives ∂z/∂s = 2(s + t)(s − t) + (s + t)². You could expand this further, but the factored form is often acceptable.

代入 x = s + t 和 y = s − t 得 ∂z/∂s = 2(s + t)(s − t) + (s + t)²。你可以进一步展开，但保留因式形式通常也可以接受。

9. Implicit Differentiation with Partial Derivatives | 用偏导数处理隐函数求导

For an equation of the form F(x, y) = 0, the derivative dy/dx can be expressed using partial derivatives:

对于形如 F(x, y) = 0 的方程，导数 dy/dx 可以用偏导数表示为：

dy/dx = − F_x / F_y, provided F_y ≠ 0

This formula is derived from the total differential dF = F_x dx + F_y dy = 0. In IB, this method is particularly useful when F is difficult or impossible to solve explicitly for y.

这个公式由全微分 dF = F_x dx + F_y dy = 0 推导而来。在IB中，当 F 难以甚至无法显式解出 y 时，这个方法特别有用。

For three variables, if F(x, y, z) = 0 defines z implicitly as a function of x and y, then:

对于三个变量，若 F(x, y, z) = 0 将 z 隐式定义为 x 和 y 的函数，则：

∂z/∂x = − F_x / F_z, and ∂z/∂y = − F_y / F_z

Be careful with signs: the minus sign is an essential part of the formula, and you must never forget it.

注意符号：公式中的负号必不可少，一定不要漏掉。

10. Applications and IB Examination Tips | 应用与IB考试提示

Partial derivatives appear in IB extensively in topics such as optimisation of functions of two variables (finding stationary points and classifying them with the second derivative test), and in the context of Lagrange multipliers if you take further Mathematics. They also appear in field theory, where quantities like electric field are related to gradients of scalar potentials.

偏导数在IB中广泛出现，例如二元函数的最优化问题（求驻点并用二阶导数判别法分类），以及如果你学习进阶数学，还会涉及拉格朗日乘子法。它们也出现在场论中，例如电场与标量势的梯度相关。

In examination questions, the most common tasks include:

在考试题目中，最常见的任务包括：

Computing first and second partial derivatives of polynomial, exponential, logarithmic, and trigonometric functions.
Evaluating a partial derivative at a specific point.
Using the chain rule to find derivatives of composite functions.
Finding the equation of a tangent plane or the gradient vector.

计算多项式、指数、对数和三角函数的一阶与二阶偏导数。
在给定点处求偏导数值。
使用链式法则求复合函数的导数。
求切平面方程或梯度向量。

To avoid mistakes, always write down which variable is held constant before differentiating. Use the notation f_x instead of ∂f/∂x if it helps you keep track. Double-check whether mixed derivatives are equal when possible.

为了避免错误，求导前要写出哪个变量保持不变。如果有助于你记住顺序，可以使用 f_x 记号代替 ∂f/∂x。在可能的情况下，检查混合导数是否相等。

11. Common Pitfalls | 常见易错点

One common mistake is to use ordinary derivative notation for a partial derivative. Another is to forget to treat constants correctly in products. For example, in f(x, y) = x sin(y), the derivative with respect to x is sin(y), not cos(y). Since y is constant with respect to x, sin(y) is just a constant factor.

一个常见错误是把偏导数写成普通导数记号。另一个常见错误是在乘积中忘记如何对待常数。例如，在 f(x, y) = x sin(y) 中，对 x 求导得到 sin(y)，而不是 cos(y)。因为 y 相对于 x 是常数，所以 sin(y) 只是一个常数因子。

Another pitfall is confusing the order in mixed partials. The expression ∂²f/∂x∂y means you differentiate with respect to y first, then x. In the notation f_xy, the subscript order tells you the opposite: first y, then x? Actually there is no universal convention, so always follow the notation defined in your textbook.

另一个易错点是混淆混合偏导的求导顺序。表达式 ∂²f/∂x∂y 表示先对 y 求导，再对 x 求导。而在记号 f_xy 中，下标的顺序有时表示先对 x 后对 y？实际上，不同教材约定不同，所以始终遵循你课本中的定义。

In IB, it is safer to write partial derivatives using fraction notation with the order of variables read from left to right in the denominator: ∂²f/∂x∂y means you differentiate in the order: first y, then x, because the derivative operator ∂/∂x is applied to ∂f/∂y. The subscript form f_xy is usually defined as (f_x)_y, meaning first x, then y. To be clear, write out the steps.

在IB中，更稳妥的做法是使用分数记号，并且按照分母中从左到右的顺序读取：∂²f/∂x∂y 表示先对 y 求导，再对 x 求导，因为算子 ∂/∂x 作用于 ∂f/∂y。下标形式 f_xy 通常定义为 (f_x)_y，即先对x后对y。为了清楚，请写出每一步。

Finally, when using the implicit function theorem, always verify that the denominator is not zero. If F_y = 0, the formula for dy/dx is undefined and you need another approach.

最后，使用隐函数定理时，务必检查分母不为零。如果 F_y = 0，则 dy/dx 的公式无定义，你需要寻找其他方法。

12. Summary | 总结

Partial derivatives allow us to study how a multivariable function changes in one coordinate direction at a time. The computation is no more difficult than ordinary differentiation, provided you keep all other variables constant. Key ideas include the limit definition, geometric meaning, higher-order and mixed partials, the chain rule, and implicit differentiation.

偏导数使我们能够分别研究多元函数在单个坐标方向上的变化。只要把其他所有变量视为常数，计算难度与普通求导相当。关键概念包括极限定义、几何意义、高阶与混合偏导、链式法则以及隐函数求导。

For IB success, practice translating between different notations, and always check whether the theorem conditions are satisfied before using a shortcut. Master these skills and you will find that partial derivatives become a reliable tool in your mathematical toolkit.

要在IB中取得好成绩，请练习在不同记号之间转换，并在使用捷径前检查定理条件是否满足。掌握这些技巧后，你会发现偏导数会成为你数学工具箱中一个可靠的工具。

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Partial Derivatives: Concepts and Computation | 偏导数的概念与计算