Parametric Differentiation: Rates, Tangents and Second Derivatives | 参数方程微分：变化率、切线与二阶导数

📚 Parametric Differentiation: Rates, Tangents and Second Derivatives | 参数方程微分：变化率、切线与二阶导数

Parametric equations describe a curve using a third variable, usually t. In Edexcel A-Level Mathematics, parametric differentiation is a core skill: it lets you find gradients, tangents, normals and second derivatives for curves that are not easily written as y = f(x). This guide covers the key methods, worked examples and common exam pitfalls.

参数方程使用第三个变量(通常是 t)来描述曲线。在 Edexcel A-Level 数学中，参数微分是一项核心技能：它可以帮助你求出不易写成 y = f(x) 的曲线的斜率、切线、法线和二阶导数。本文涵盖关键方法、例题和常见考试失分点。

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

A parametric curve is defined by a pair of equations x = f(t), y = g(t), where t is the parameter. As t varies, the point (x, y) traces out a curve. This representation is especially useful for circles, ellipses, cycloids and motion problems.

参数曲线由一对方程 x = f(t)、y = g(t) 定义，其中 t 是参数。当 t 变化时，点 (x, y) 描绘出一条曲线。这种表示方式对于圆、椭圆、摆线以及运动问题尤其有用。

• Circle: x = r cos t, y = r sin t | 圆：x = r cos t, y = r sin t
• Parabola: x = at², y = 2at | 抛物线：x = at², y = 2at
• Line: x = x₀ + at, y = y₀ + bt | 直线：x = x₀ + at, y = y₀ + bt

中文：参数 t 可以表示角度、时间或任意独立变量。理解参数如何改变点的位置，是处理参数曲线的基础。

2. The Chain Rule Link | 链式法则的联系

The key idea is that y and x both depend on t, so we can link dy/dx to dy/dt and dx/dt using the chain rule. From dy/dt = dy/dx × dx/dt, we obtain the central formula:

关键思想是 y 和 x 都依赖于 t，因此我们可以用链式法则把 dy/dx 与 dy/dt、dx/dt 联系起来。由 dy/dt = dy/dx × dx/dt，可得核心公式：

dy/dx = (dy/dt) ÷ (dx/dt)

This formula is valid only when dx/dt ≠ 0. If dx/dt = 0 at a point, the tangent is vertical, so the gradient is undefined.

该公式仅在

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