Parametric Differentiation | 参数方程求导

📚 Parametric Differentiation | 参数方程求导

Parametric differentiation is a core technique in Edexcel A-Level Mathematics. It allows you to find the gradient of a curve when x and y are both defined in terms of a third variable, usually t or θ.

参数方程求导是 Edexcel A-Level 数学的核心技巧。当 x 和 y 都由第三个变量（通常是 t 或 θ）定义时，它可以用来求曲线的梯度。

1. Parametric Equations Overview | 参数方程概述

A parametric curve is defined by a pair of equations such as x = f(t) and y = g(t). As the parameter t changes, the point (x, y) traces out a curve in the xy-plane.

参数曲线由一对方程定义，例如 x = f(t) 和 y = g(t)。当参数 t 变化时，点 (x, y) 在 xy 平面上描绘出一条曲线。

For example, x = t² and y = 2t represents a parabola. Some curves, such as circles and cycloids, are much easier to describe parametrically than in the form y = f(x).

例如，x = t² 和 y = 2t 表示一条抛物线。有些曲线，例如圆和摆线，用参数形式描述要比 y = f(x) 形式容易得多。

x = a cos θ, y = a sin θ — a circle of radius a — 半径为 a 的圆
x = at², y = 2at — a parabola — 抛物线
x = r(θ − sin θ), y = r(1 − cos θ) — a cycloid — 摆线

The parameter often represents time or an angle, which makes parametric equations very natural for modelling motion.

参数通常代表时间或角度，这使得参数方程在描述运动时非常自然。

2. Why Use Parametric Differentiation? | 为什么要使用参数求导？

Many important curves cannot be written as a single function y = f(x). A circle is a classic example, because it fails the vertical line test.

许多重要曲线无法写成单一函数 y = f(x)。圆就是一个典型例子，因为它不满足竖直直线检验。

In mechanics, position is often given by x = f(t) and y = g(t). To find the gradient or the direction of motion, you need dy/dx, and that is exactly what parametric differentiation provides.

在力学中，位置通常由 x = f(t) 和 y = g(t) 给出。为了求梯度或运动方向，你需要 dy/dx，而这正是参数方程求导所提供的结果。

Edexcel exam questions usually expect you to differentiate parametrically rather than eliminating the parameter, because elimination can be slow or algebraically messy.

Edexcel 考试题通常希望你对参数方程直接求导，而不是消去参数，因为消元可能很慢，或者代数运算非常复杂。

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

The rule for parametric differentiation comes directly from the chain rule. If y and x are both functions of t, then:

参数求导法则直接来自链式法则。如果 y 和 x 都是 t 的函数，那么：

dy/dt = dy/dx × dx/dt

Rearranging this gives the key formula used in nearly all parametric differentiation questions.

重新整理后得到几乎所有参数方程求导题目都会用到的关键公式。

dy/dx = (dy/dt) ÷ (dx/dt) = (dy/dt) / (dx/dt)

This formula is valid only where dx/dt ≠ 0. If dx/dt = 0, the tangent is vertical or the derivative is undefined.

该公式仅在 dx/dt ≠ 0 时成立。如果 dx/dt = 0，则切线是竖直的，或者导数不存在。

In words: differentiate y with respect to t, differentiate x with respect to t, then divide the first result by the second.

用文字表述：先对 y 关于 t 求导，再对 x 关于 t 求导，然后用第一个结果除以第二个结果。

A useful memory tip is “dy over dx equals dy over dt over dx over dt”. Do not reverse the division order.

一个有用的记忆方法是：“dy/dx 等于 dy/dt 除以 dx/dt”。不要把除法的顺序反过来。

4. Finding dy/dx Step by Step | 逐步求 dy/dx

To find dy/dx from parametric equations, follow these steps carefully.

要从参数方程求 dy/dx，请仔细遵循以下步骤。

Step 1: Compute dx/dt — 步骤 1：计算 dx/dt。
Step 2: Compute dy/dt — 步骤 2：计算 dy/dt。
Step 3: Write dy/dx = (dy/dt) / (dx/dt) — 步骤 3：写出 dy/dx = (dy/dt) / (dx/dt)。
Step 4: Simplify your answer if possible — 步骤 4：尽可能化简答案。

For a quick example, if x = 2t and y = t², then dx/dt = 2 and dy/dt = 2t, so dy/dx = 2t / 2 = t.

举一个快速例子，如果 x = 2t 且 y = t²，那么 dx/dt = 2，dy/dt = 2t，所以 dy/dx = 2t / 2 = t。

Always check whether your final expression can be simplified, as this often makes later parts of a question easier.

一定要检查最终表达式是否可以化简，因为这通常会让题目后面的部分更容易处理。

5. Worked Example 1 | 例题 1

A curve is defined by x = t² + 1 and y = t³ + t. Find dy/dx in terms of t.

一条曲线由 x = t² + 1 和 y = t³ + t 定义。求用 t 表示的 dy/dx。

First compute dx/dt = 2t and dy/dt = 3t² + 1.

首先计算 dx/dt = 2t，dy/dt = 3t² + 1。

Then, using the formula:

然后使用公式：

dy/dx = (3t² + 1) / (2t)

This expression gives the gradient at any point where t ≠ 0. For example, at t = 2, the gradient is (3 × 4 + 1) / (4) = 13/4.

这个表达式给出了除 t = 0 以外任意点处的梯度。例如，在 t = 2 处，梯度为 (3 × 4 + 1) / 4 = 13/4。

6. Second Derivative | 二阶导数

The second derivative d²y/dx² measures the rate of change of the gradient. It is not obtained by differentiating dy/dx with respect to

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Parametric Differentiation | 参数方程求导