Differentiation of Parametric Equations | 参数方程求导

📚 Differentiation of Parametric Equations | 参数方程求导

In Edexcel A-Level Mathematics, parametric equations express the coordinates of points on a curve using a third variable, usually t or θ. To analyse the gradient, tangents, and normal lines, you must differentiate these parametric forms correctly. This article guides you through the core techniques for finding dy/dx and d²y/dx² when x and y are defined in terms of a parameter.

在 Edexcel A-Level 数学中，参数方程用一个第三变量（通常是 t 或 θ）来表示曲线上点的坐标。要分析梯度、切线和法线，就必须正确地对这些参数形式进行求导。本文将引导你掌握当 x 和 y 由参数定义时，求解 dy/dx 和 d²y/dx² 的核心技巧。

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1. What Are Parametric Equations? | 什么是参数方程？

circle can be expressed as x = cos θ, y = sin θ. This representation is particularly useful for curves that double back on themselves or involve complex motion.

参数方程不是直接将 y 写成 x 的函数，而是用一个独立的参数同时定义 x 和 y。例如，单位圆可以表示为 x = cos θ, y = sin θ。这种表示法对于自交曲线或涉及复杂运动的曲线尤为有用。

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2. Why Standard Differentiation Fails | 为什么标准求导法不适用

ectly. With parametric equations, y is not explicitly given as a function of x, so you cannot differentiate y with respect to x immediately. You need an intermediate step that relates the rates of change with respect to the parameter.

如果你有 y = f(x)，可以直接求得 dy/dx。而在参数方程中，y 并未显式地表示为 x 的函数，因此无法立即对 x 求导。你需要一个中间步骤，将变化率与参数联系起来。

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3. The Chain Rule Connection | 链式法则的联系

se dy/dt = (dy/dx) · (dx/dt), and rearranging gives the required derivative. Always check that the denominator is not zero to avoid undefined gradients at that point.

参数求导的关键恒等式来源于链式法则：dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。因为 dy/dt = (dy/dx) · (dx/dt)，重新整理即可得到所需的导数。务必要检查分母是否为零，以避免在该点出现未定义的梯度。

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4. Step-by-Step Procedure for dy/dx | 求 dy/dx 的分步步骤

It is good practice to simplify the result, cancelling any common factors of t where possible. For trigonometric forms, use identities like sec² t = 1 + tan² t to tidy up.

首先，对 t 求导 x 得到 dx/dt。其次，对 t 求导 y 得到 dy/dt。最后，用 dy/dt 除以 dx/dt。最好化简结果，尽可能消去 t 的公因式。对于三角函数式，可利用恒等式如 sec² t = 1 + tan² t 进行整理。

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5. Worked Example: Polynomial Parameter | 示例：多项式参数

Given x = t³ − 3t, y = t² + 2t, find dy/dx in terms of t.
dx/dt = 3t² − 3, dy/dt = 2t + 2.
Thus dy/dx = (2t + 2) / (3t² − 3) = [2(t+1)] / [3(t²−1)].
actorising further: t²−1 = (t−1)(t+1), so dy/dx = 2 / [3(t−1)], provided t ≠ −1 and t ≠ 1.

已知 x = t³ − 3t, y = t² + 2t，求用 t 表示的 dy/dx。
dx/dt = 3t² − 3，dy/dt = 2t + 2。
因此 dy/dx = (2t + 2) / (3t² − 3) = [2(t+1)] / [3(t²−1)]。
进一步因式分解：t²−1 = (t−1)(t+1)，所以 dy/dx = 2 / [3(t−1)]，前提是 t ≠ −1 且 t ≠ 1。

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6. Worked Example: Trigonometric Parameter | 示例：三角函数参数

ngle identity cos 2θ = 1 − 2 sin² θ, you can rewrite the derivative as −(1 − 2 sin² θ)/sin θ = 2 sin θ − csc θ.

一条曲线由 x = 2 cos θ, y = sin 2θ 定义。首先，dx/dθ = −2 sin θ, dy/dθ = 2 cos 2θ。然后 dy/dx = (2 cos 2θ) / (−2 sin θ) = −cos 2θ / sin θ。利用倍角恒等式 cos 2θ = 1 − 2 sin² θ，可将导数改写为 −(1 − 2 sin² θ)/sin θ = 2 sin θ − csc θ。

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7. Finding the Second Derivative d²y/dx² | 求二阶导数 d²y/dx²

then dividing by dx/dt: d²y/dx² = d(dy/dx)/dt ÷ dx/dt. Never simply differentiate dy/dx with respect to x without accounting for the parameter.

二阶导数度量了梯度的变化率。在参数形式下，先对 t 求导 dy/dx，再除以 dx/dt 来得到二阶导数：d²y/dx² = d(dy/dx)/dt ÷ dx/dt。绝不能忽略参数而直接对 x 求导 dy/dx。

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8. Applying the Second Derivative Formula | 二阶导数公式的应用

en recall dx/dt = 3(t²−1) = 3(t−1)(t+1). Therefore d²y/dx² = [−(2/3)(t−1)⁻²] ÷ [3(t−1)(t+1)] = −2 / [9 (t−1)³ (t+1)].

沿用第5节的多项式示例，dy/dx = 2/[3(t−1)] = (2/3)(t−1)⁻¹。对 t 求导：d(dy/dx)/dt = −(2/3)(t−1)⁻²。再由 dx/dt = 3(t²−1) = 3(t−1)(t+1)。因此 d²y/dx² = [−(2/3)(t−1)⁻²] ÷ [3(t−1)(t+1)] = −2 / [9 (t−1)³ (t+1)]。

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9. Equations of Tangents and Normals | 切线和法线方程

e dy/dx is known at a specific parameter value t₀, the tangent line has gradient m = dy/dx| titute the parameter into the original parametric equations to find the point coordinates.

一旦在特定参数值 t₀ 下求得 dy/dx，切线梯度为 m = dy/dx|t₀。其方程为 y − y(t₀) = m (x − x(t₀))。法线与之垂直，故其梯度为 −1/m。务必先将参数代入原参数方程求出点的坐标。

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10. Stationary Points via Parametric Curves | 参数曲线的驻点

ile ensuring dx/dt ≠ 0 locates the parameter values for turning points. After finding t, determine x and y, then check the sign of d²y/dx² to classify maxima, minima, or points of inflection.

驻点出现在 dy/dx = 0 处。令分子 dy/dt 等于零，同时确保 dx/dt ≠ 0，即可找到对应拐点的参数值。求出 t 后，计算 x 和 y，再通过 d²y/dx² 的正负来区分为极大值、极小值或拐点。

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11. Common Pitfalls and How to Avoid Them | 常见错误与避免方法

• Forgetting the chain rule and trying to differentiate y with respect to x directly.
• Dividing in the wrong order: dy/dx is dy/dt divided by dx/dt, not the reverse.
• gnoring domain restrictions where dx/dt = 0, which can lead to vertical tangents or cusps.
• Mishandling negative signs, especially in trigonometric differentiation.
• Omitting simplification, making the second derivative unnecessarily messy.

• 忘记链式法则，试图直接对 x 求导 y。
• 除反了：dy/dx 是 dy/dt 除以 dx/dt，而非相反。
• 忽略 dx/dt = 0 的定义域限制，这可能导致垂直切线或尖点。
• 处理负号不当，尤其在三角求导中。
• 省略化简步骤，导致二阶导数表达式不必要地繁杂。

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12. Exam Strategy and Summary | 考试策略与总结

uation, or verify a given derivative. Write your working clearly, show the division step, and always state when dx/dt = 0 would cause an invalid derivative. Practise converting parametric forms into Cartesian equations only when needed for checking, not as a primary method for differentiation.

在 Edexcel 考试中，参数求导题常出现在纯数学试卷二中。你可能需要求梯度、切线方程，或验证给定的导数。清晰地写出运算过程，展示除法步骤，并始终注明何时 dx/dt = 0 会导致导数无效。仅在需要检验时才将参数形式转化为笛卡尔方程，不要将其作为主要的求导方法。

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