Exercise 6F.2: Implicit and Parametric Differentiation | 习题 6F.2：隐函数与参数微分

📚 Exercise 6F.2: Implicit and Parametric Differentiation | 习题 6F.2：隐函数与参数微分

). These include implicit relations and parametrically defined curves. Mastery of these techniques is essential for working with related rates, finding tangents to circles, ellipses, and other curves not given as explicit functions.

在IB数学分析与方法课程中，习题6F.2侧重于对那些难以写成 y = f(x) 形式的方程进行求导。这包括隐式关系和参数定义的曲线。掌握这些技巧对于处理相关变化率、求圆、椭圆以及其他非显函数曲线的切线至关重要。

1. Why Implicit Differentiation? | 为什么需要隐函数求导？

is given by an equation like x² + y² = 25, solving for y gives two explicit functions: y = √25 − x² and y = −√25 − x². Differentiating each piece separately is messy. Implicit differentiation allows us to find dy/dx directly from the original equation without solving for y.

当一个包含 x 和 y 的关系由方程给出时，例如 x² + y² = 25，解出 y 会得到两个显函数：y = √25 − x² 和 y = −√25 − x²。对每一部分分别求导很麻烦。隐函数求导使我们无需解出 y 就能直接从原方程求出 dy/dx。

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2. The Core Mechanism: Chain Rule Every Time | 核心机制：时刻使用链式法则

d/dx(y²) = 2y · dy/dx. This is simply the chain rule with an unknown inner function y(x).

关键在于把 y 视为 x 的隐函数，每当我们对含有 y 的项求导时都要使用链式法则。例如，d/dx(y²) = 2y · dy/dx。这只是对未知内层函数 y(x) 应用了链式法则。

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3. Step-by-Step Example: Circle Equation | 逐步示例：圆的方程

Take x² + y² = 25. Differentiate both sides with respect to x:
d/dx(x²) + d/dx(y²) = d/dx(25)
2x + 2y (dy/dx) = 0.
Now solve for dy/dx: 2y (dy/dx) = −2x, so dy/dx = −x/y, provided y ≠ 0.

以 x² + y² = 25 为例。对两边关于 x 求导：
d/dx(x²) + d/dx(y²) = d/dx(25)
2x + 2y (dy/dx) = 0。
现在解出 dy/dx：2y (dy/dx) = −2x，所以 dy/dx = −x/y，前提是 y ≠ 0。

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4. Handling Product Terms Involving x and y | 处理含有 x 与 y 的乘积项

r with the chain rule. For instance, d/dx(xy) = x · d/dx(y) + y · d/dx(x) = x (dy/dx) + y.

如果方程包含像 xy 这样的项，我们必须同时使用乘法法则和链式法则。例如，d/dx(xy) = x · d/dx(y) + y · d/dx(x) = x (dy/dx) + y。

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5. Worked Example: xy + sin y = 1 | 工作示例：xy + sin y = 1

Differentiate both sides of xy + sin y = 1:
d/dx(xy) + d/dx(sin y) = 0
(x dy/dx + y) + cos y · dy/dx = 0.
Collect dy/dx terms: dy/dx (x + cos y) = −y, so dy/dx = −y/(x + cos y).

对 xy + sin y = 1 两边求导：
d/dx(xy) + d/dx(sin y) = 0
(x dy/dx + y) + cos y · dy/dx = 0。
合并 dy/dx 项：dy/dx (x + cos y) = −y，所以 dy/dx = −y/(x + cos y)。

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6. Parametric Equations: The Basic Idea | 参数方程：基本思想

y are functions of t, the chain rule gives dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

当一条曲线由参数定义时，例如 x = f(t)，y = g(t)，我们经常需要求 dy/dx。由于 x 和 y 都是 t 的函数，由链式法则可得 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。

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7. Worked Example: Parametric Differentiation | 工作示例：参数求导

Given x = t² + 1, y = 2t − t³, find dy/dx at t = 1.
Compute dx/dt = 2t and dy/dt = 2 − 3t².
Then dy/dx = (2 − 3t²)/(2t).
At t = 1, dy/dx = (2 − 3)/2 = −1/2.

已知 x = t² + 1，y = 2t − t³，求在 t = 1 处的 dy/dx。
计算 dx/dt = 2t，dy/dt = 2 − 3t²。
则 dy/dx = (2 − 3t²)/(2t)。
在 t = 1 处，dy/dx = (2 − 3)/2 = −1/2。

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8. Second Derivatives in Parametric Form | 参数形式的二阶导数

, dy/dx = (2 − 3t²)/(2t). Differentiate this with respect to t, then divide by dx/dt = 2t.

二阶导数 d²y/dx² 并不是简单地把 d²y/dt² 除以 d²x/dt²。应使用公式 d²y/dx² = d/dx(dy/dx) = [d/dt(dy/dx)] / (dx/dt)。以前一题为例，dy/dx = (2 − 3t²)/(2t)。将其对 t 求导，再除以 dx/dt = 2t。

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9. Implicit Second Derivative Example | 隐函数二阶导数示例

y/dx)/y². Substitute dy/dx = −x/y to simplify and obtain d²y/dx² = −(y² + x²)/y³ = −25/y³.

对于 x² + y² = 25，我们已经求出 dy/dx = −x/y。为得到 d²y/dx²，用商法则对 dy/dx 关于 x 求导：d/dx(−x/y) = −(y · 1 − x · dy/dx)/y²。代入 dy/dx = −x/y 简化，得到 d²y/dx² = −(y² + x²)/y³ = −25/y³。

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10. Common Pitfalls and Tips | 常见错误与提示

Forgetting to apply the chain rule to y-terms is the most frequent mistake. Always write dy/dx immediately after differentiating a y-factor. In parametric problems, ensure dx/dt is not zero at the point of interest to avoid division by zero. When finding tangents at a given point, you can substitute the point's coordinates into the derivative expression.

忘记对含 y 的项使用链式法则是最常见的错误。始终在对 y 因子求导后立即写出 dy/dx。在参数问题中，确保在关注点处 dx/dt 不为零，以避免除以零。在给定点处求切线时，可将点的坐标代入导数表达式。

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11. Connecting Implicit Differentiation to Related Rates | 隐函数求导与相关变化率的联系

+ y² = 25 gives 2x(dx/dt) + 2y(dy/dt) = 0, which can be rearranged to find unknown rates.

关于时间 t 的隐函数求导是相关变化率问题的基础。如果 x 和 y 都是时间的函数，对 x² + y² = 25 这样的方程求导得到 2x(dx/dt) + 2y(dy/dt) = 0，可以重新整理求出未知的变化率。

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12. IB Exam-Style Question | IB 考试风格题目

2, dy/dt = 2 cos(π/3) = 1, so dy/dx = 1 / (−1/2) = −2. The point is (cos π/6, sin π/3) = (√3/2, √3/2). Tangent: y − √3/2 = −2(x − √3/2).

一条曲线由参数方程 x = cos t, y = sin 2t 定义。求在 t = π/6 处的切线方程。首先，dx/dt = −sin t，dy/dt = 2 cos 2t。在 t = π/6 处，dx/dt = −1/2，dy/dt = 2 cos(π/3) = 1，因此 dy/dx = 1 / (−1/2) = −2。点为 (cos π/6, sin π/3) = (√3/2, √3/2)。切线：y − √3/2 = −2(x − √3/2)。

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