📚 6. An Extension of the Chain Rule | 6. 链式法则扩展
nalysis and Approaches, students must master several important extensions: parametric differentiation, implicit differentiation, related rates, higher-order chain rule applications, and even the chain rule for functions of several variables. Each of these topics relies on the same core principle — dy/dx = dy/du · du/dx — ons, complete with examples, common pitfalls, and connections between the different forms.
链式法则是微积分中最灵活的工具之一，其威力远不止于 y = sin(3x) 这样的简单复合函数。在 IB 数学分析与方法课程中，学生必须掌握几个重要的扩展：参数方程求导、隐函数求导、相关变化率、高阶链式法则以及多变量函数的链式法则。这些主题都基于同一个核心原理 —— dy/dx = dy/du · du/dx——但将其应用到越来越复杂的情境中。本文全面回顾这些扩展，辅以例题、常见错误以及不同形式之间的联系。
1. Review of the Basic Chain Rule | 基本链式法则回顾
y with respect to x is given by dy/dx = dy/du · du/dx. In function notation, this reads (f ∘ g)'(x) = f'(g(x)) · g'(x). The Leibniz notation highlights the 'cancellation' x = 5(2x+1)⁴·2 = 10(2x+1)⁴. This basic rule is the foundation for everything that follows.
在探索扩展之前，有必要回顾链式法则的基本形式。如果 y = f(u) 与 u = g(x) 都是可微函数，那么 y 对 x 的导数为 dy/dx = dy/du · du/dx。用函数记号表示为 (f ∘ g)'(x) = f'(g(x)) · g'(x)。莱布尼茨记号突显了“约分”直觉，即 du 似乎像分数一样被约去，尽管严格证明依赖于极限。一个简单例子：对于 y = (2x + 1)⁵，令 u = 2x + 1，于是 y = u⁵。那么 dy/du = 5u⁴，du/dx = 2，得到 dy/dx = 5(2x+1)⁴·2 = 10(2x+1)⁴。这一基本规则是后续所有内容的基础。
2. Parametric Differentiation | 参数方程求导
te variable t connects x and y. For instance, if x = t² + 1 and y = t³ − t, then dx/dt = 2t, dy/dt = 3t² − 1, so dy/dx = (3t² − 1)/(2t). Notice that the answer is expressed in terms of the parameter t, which is typical for parametrics.
当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，我们经常需要梯度 dy/dx。链式法则提供了一条直接途径：由于 dy/dt = dy/dx · dx/dt，只要 dx/dt ≠ 0，便可重排得到 dy/dx = (dy/dt) / (dx/dt)。这是链式法则的扩展，中间变量 t 连接了 x 和 y。例如，若 x = t² + 1，y = t³ − t，则 dx/dt = 2t，dy/dt = 3t² − 1，因此 dy/dx = (3t² − 1)/(2t)。注意答案用参数 t 表示，这正是参数方程求导的典型特征。
(4t³). This shows how repeated use of the chain rule handles higher-order derivatives elegantly.
对于参数方程的二阶导数 d²y/dx²，我们再次运用链式法则：d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt)。在上例中，dy/dx = (3t² − 1)/(2t) = (3/2)t − (1/2)t⁻¹，因此 d/dt (dy/dx) = 3/2 + (1/2)t⁻² = (3t² + 1)/(2t²)。除以 dx/dt = 2t 得到 d²y/dx² = (3t² + 1)/(4t³)。这表明反复使用链式法则可优雅地处理高阶导数。
3. Implicit Differentiation | 隐函数求导
and applying the chain rule to terms involving y gives: d/dx (x²) + d/dx (y²) = d/dx (25) ⇒ that d/dx (y²) = 2y·dy/dx, exactly as the chain rule prescribes for a composite function.
隐函数求导是一项强大技巧，用于当 y 由形如 x² + y² = 25 的方程隐式定义时求出 dy/dx。对方程两边关于 x 求导，并对含 y 的项应用链式法则：d/dx (x²) + d/dx (y²) = d/dx (25) ⇒ 2x + 2y · dy/dx = 0。解得 dy/dx = −x/y。关键步骤在于将 y 视为 x 的函数，从而 d/dx (y²) = 2y·dy/dx，这正是链式法则对复合函数的要求。
th the direct method. This formula is particularly useful for complicated implicit relations where solving for y is impractical.
更一般地，若方程 F(x, y) = 0 隐式定义了 y，多变量链式法则（稍后介绍）给出 dF/dx = ∂F/∂x + ∂F/∂y · dy/dx = 0，因此 dy/dx = −(∂F/∂x)/(∂F/∂y)。对于圆，F(x,y) = x² + y² − 25，得到 dy/dx = −(2x)/(2y) = −x/y，与直接方法一致。当隐式关系复杂、难以解出 y 时，这一公式尤其有用。
4. Related Rates | 相关变化率
a constant rate dr/dt = 2 cm/s. How fast is the volume increasing when the radius is 10 cm? ³/s. The chain rule links the rates dV/dt and dr/dt through the ordinary derivative dV/dr.
相关变化率问题通过将一个量的变化率与已知变化率的其他量建立联系，从而求出该量的变化率。链式法则是这类问题背后的引擎。例如，一个球形气球被充气，使其半径以恒定速率 dr/dt = 2 cm/s 增加。当半径为 10 cm 时，体积增加有多快？由于 V = (4/3)πr³，对 t 求导：dV/dt = dV/dr · dr/dt = 4πr² · dr/dt。代入 r = 10 和 dr/dt = 2 得到 dV/dt = 4π(100)·2 = 800π cm³/s。链式法则通过普通导数 dV/dr 将变化率 dV/dt 和 dr/dt 联系起来。
and h is V = 5h² (if the triangle is isosceles and base equals height). Then dV/dt = dV/dh · dh/dt ⇒ illustrates the broad applicability of the chain rule in modelling real-world situations.
在更复杂的情形中，多个变量通过方程联系在一起，必须对每个变量应用链式法则。设想一个长 10 m、截面为三角形的水槽，水以 0.5 m³/min 的速率流入。体积 V 依赖于水位高度 h 和长度，且 V 与 h 的关系为 V = 5h²（假设三角形为等腰且底等于高）。那么 dV/dt = dV/dh · dh/dt ⇒ 10h · dh/dt。已知 dV/dt = 0.5，便可求出 dh/dt。这显示了链式法则在建模实际情境中的广泛适用性。
5. Higher-Order Chain Rule | 高阶链式法则
For twice-differentiable composites y = f(g(x)), the first derivative is y' = f'(g(x))·g' e second derivative requires the product rule and another application of the chain rule: y'' = f''(g(x))·[g'(x)]² + f'(g(x))·g'' ) = d²y/du² · (du/dx)² + dy/du · d²u/dx². This shows that second derivatives do not simply “chain” like first derivatives; an extra term appears due to the derivative of du/dx itself.
对于二次可微的复合函数 y = f(g(x))，一阶导数为 y' = f'(g(x))·g'(x)。二阶导数需要乘积法则并再次应用链式法则：y'' = f''(g(x))·[g'(x)]² + f'(g(x))·g''(x)。用莱布尼茨记号表示，若 y = f(u) 且 u = g(x)，则 d²y/dx² = d/dx (dy/du · du/dx) = d²y/du² · (du/dx)² + dy/du · d²u/dx²。这表明二阶导数并非像一阶导数那样简单地“链式相乘”；由于 du/dx 本身的导数，会增加一个额外项。
As an example, take y = sin(x²). Here f(u) = sin u, g(x) = x², so f'(u) = cos u, f''(u) = −sin u, g'(x) = 2x, g''(x) = 2. Then y'' = −sin(x²)·(2x)² + cos(x²)·2 = −4x² sin(x²) + 2 cos(x²). Students often forget the [g' stake. This higher-order chain rule is essential for Taylor series and curvature problems.
以 y = sin(x²) 为例。这里 f(u) = sin u，g(x) = x²，故 f'(u) = cos u，f''(u) = −sin u，g'(x) = 2x，g''(x) = 2。于是 y'' = −sin(x²)·(2x)² + cos(x²)·2 = −4x² sin(x²) + 2 cos(x²)。学生常遗漏 [g'(x)]² 这个因子；刻意练习公式有助于避免这一错误。这一高阶链式法则对于泰勒级数和曲率问题至关重要。
6. The Chain Rule for Functions of Several Variables | 多变量函数的链式法则
here each branch contributes a term. For instance, let z = x²y, with x = cos t and y = eᵗ. Then ∂z/∂x = 2xy, ∂z/∂y = x², dx/dt = −sin t, dy/dt = eᵗ. Hence dz/dt = 2xy·(−sin t) + x²·eᵗ = −2 cos t · eᵗ · sin t + cos²t · eᵗ, which simplifies to eᵗ cos t (cos t − 2 sin t) after substitution.
当函数依赖于多个自身又依赖于另一变量的变量时，链式法则会自然推广。若 z = f(x, y)，且 x = x(t)，y = y(t)，则 z 对 t 的全导数为 dz/dt = ∂z/∂x · dx/dt + ∂z/∂y · dy/dt。这可用树状图可视化，其中每条分支贡献一项。例如，设 z = x²y，x = cos t，y = eᵗ。那么 ∂z/∂x = 2xy，∂z/∂y = x²，dx/dt = −sin t，dy/dt = eᵗ。因此 dz/dt = 2xy·(−sin t) + x²·eᵗ = −2 cos t · eᵗ · sin t + cos²t · eᵗ，代入后化简为 eᵗ cos t (cos t − 2 sin t)。
ral to implicit differentiation (as seen earlier) and to the study of partial differential equations. In IB, it appears in optimisation and rates of change involving multiple constraints.
如果中间变量本身依赖于两个自变量，比如 x = x(u, v) 且 y = y(u, v)，那么 z 对 u 和 v 的偏导数由下式给出：∂z/∂u = ∂z/∂x · ∂x/∂u + ∂z/∂y · ∂y/∂u，∂z/∂v 同理。这一扩展是隐函数求导（如前文所见）以及偏微分方程学习的核心。在 IB 课程中，它出现在涉及多重约束的优化和变化率问题中。
7. Application in Differential Equations | 在微分方程中的应用
s is validated by the chain rule in reverse: if y is a function of x, then d/dx [F(y)] = F'(y) dy/dx, which means ∫ F'(y) dy/dx dx = ∫ F'(y) dy. By setting F' Thus, the technique of separation of variables is a direct consequence of the chain rule.
形如 dy/dx = g(x)h(y) 的可分离一阶微分方程可通过改写为 (1/h(y)) dy = g(x) dx 并两边积分求解。这一过程由链式法则反向保证：若 y 是 x 的函数，则 d/dx [F(y)] = F'(y) dy/dx，这意味着 ∫ F'(y) dy/dx dx = ∫ F'(y) dy。令 F'(y) = 1/h(y)，积分 ∫ (1/h(y)) dy 便有了意义。因此，分离变量法正是链式法则的直接推论。
egration by substitution, or u-substitution, also rests on the chain rule. For ∫ f(g(x)) g'(x) dx, letting u = g(x) gives du = g' y of the chain rule in both differentiation and integration modes is indispensable for tackling IB calculus questions involving rates, areas, and volumes.
换元积分法（或 u-代换）同样基于链式法则。对于 ∫ f(g(x)) g'(x) dx，令 u = g(x) 得到 du = g'(x) dx，积分便化为 ∫ f(u) du。识别内部函数的导数至关重要，做不到是常见错误。熟练运用链式法则的求导与积分两种模式，对于解决 IB 微积分中涉及速率、面积和体积的问题不可或缺。
8. Common Mistakes and Tips | 常见错误与技巧
take is omitting the derivative of the inner function, especially in composites like y = e2x where students might write y' = e2x instead of 2e2x. Always ask: “What is the derivative of the inside?” A checklist approach — identify outer function, inner function, differentiate each, multiply — , e.g. d/dx (y³) = 3y², not 3y² dy/dx. The latter is correct because y is a function of x.
一个常见错误是遗漏内部函数的导数，特别是在 y = e2x 这类复合函数中，学生可能写成 y' = e2x 而非 2e2x。始终要问自己：“内部函数的导数是什么？”采用清单式方法——识别外部函数、内部函数，分别求导，相乘——可减少错误。隐函数求导中还有一个陷阱：对 y 的函数求导时忘记乘 dy/dx，例如误以为 d/dx (y³) = 3y²，而不是 3y² dy/dx。后者才是正确的，因为 y 是 x 的函数。
/ (d²x/dt²), which is false. The correct formula involves differentiating dy/dx as a function of t and then dividing by dx/dt. A helpful mnemonic: “second derivative is the derivative of the first derivative, with respect to x, not t.” ng a diagram and writing down known rates symbolically before differentiating can clarify the relationships.
在参数方程中，学生有时在求 d²y/dx² 时错误地约去 dt，写成 d²y/dx² = (d²y/dt²) / (d²x/dt²)，这是错误的。正确公式涉及将 dy/dx 视为 t 的函数求导，再除以 dx/dt。一个助记方法是：“二阶导数是 first derivative 对 x 的导数，而不是对 t。”最后，在处理相关变化率时，谨慎处理单位和一致的符号可防止计算错误。在求导之前画出关系图并用符号写下已知速率，可理清各量之间的关联。
9. Proving the Chain Rule Extensions | 链式法则扩展的证明思路
gives dy/dx = (dy/dt)/(dx/dt), assuming dx/dt ≠ 0. This is justified by the inverse function rule and continuity. For the multivariable chain rule, the idea is that a small change Δt causes changes in x and y, which together change z: Δz ≈ (∂z/∂x)Δx + (∂z/∂y)Δy. Dividing by Δt and taking limits yields the sum of partial derivatives times ordinary derivatives, provided the function is differentiable.
尽管 IB 不要求严格的 ε-δ 证明，但理解扩展背后的逻辑可加深理解。参数方程求导的证明从 dy/dt = dy/dx · dx/dt 开始，这无非是将基本链式法则应用于中间变量 x(t)。重排后可得到 dy/dx = (dy/dt)/(dx/dt)，前提是 dx/dt ≠ 0。这一步由反函数法则和连续性保证。对于多变量链式法则，其思想是：微小变化 Δt 引起 x 和 y 的变化，二者共同改变 z：Δz ≈ (∂z/∂x)Δx + (∂z/∂y)Δy。两边除以 Δt 并
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