📚 A-Level Maths: Differentiation of Parametric Equations | A-Level 数学：参数方程的求导方法
Parametric equations are a powerful way to describe curves in A-Level mathematics. Instead of writing y directly as a function of x, we write both x and y in terms of a third variable, usually t. Differentiation then allows us to find the gradient, the equation of tangents and normals, and the location and nature of stationary points. This article will guide you through the method step by step.
参数方程是 A-Level 数学中描述曲线的有力工具。我们不再把 y 直接写成 x 的函数，而是把 x 和 y 都写为第三个变量（通常是 t）的函数。通过求导，我们可以得到曲线的斜率、切线与法线方程，以及驻点的位置和性质。本文将逐步讲解参数方程求导的方法。
1. What Are Parametric Equations? | 什么是参数方程？
l for curves that cannot be written easily in the form y = f(x), such as circles or loops.
在参数形式中，一条曲线由两个方程定义：x = f(t) 和 y = g(t)。变量 t 称为参数，每一个 t 的取值对应曲线上的一个点 (x, y)。这种方法对于无法方便写成 y = f(x) 的曲线（例如圆或环线）特别有用。
example is x = t, y = t², which is simply the parabola y = x² written in parametric form.
例如，半径为 1 的圆可以写成 x = cos θ，y = sin θ，其中 θ 是参数。另一个常见例子是 x = t，y = t²，这其实就是抛物线 y = x² 的参数形式。
| Cartesian form | Parametric form |
| y = x² | x = t, y = t² |
| x² + y² = 1 | x = cos θ, y = sin θ |
| y = x² - 2x | x = t, y = t² - 2t |
The parameter can also represent time, angle, or another physical quantity. In many exam questions, t and θ are the two most common parameters. The key skill is finding dy/dx without eliminating the parameter.
参数也可以表示时间、角度或其他物理量。在许多考试题中，t 和 θ 是最常见的两个参数。关键技能是在不消去参数的情况下求出 dy/dx。
2. The Key Formula: dy/dx = (dy/dt) ÷ (dx/dt) | 核心公式：dy/dx = (dy/dt) ÷ (dx/dt)
Suppose x and y are both functions of t. To find dy/dx, we use the chain rule. Since x depends on t, we can write the derivative as a product of dy/dt and dt/dx.
假设 x 和 y 都是 t 的函数。为了求出 dy/dx，我们使用链式法则。因为 x 依赖于 t，我们可以把导数写成 dy/dt 与 dt/dx 的乘积。
dy/dx = dy/dt × dt/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0
nator is zero and the tangent may be vertical. If both dx/dt and dy/dt are zero, the point is called a singular point and the gradient is not uniquely defined by this formula.
条件 dx/dt ≠ 0 非常重要。如果在某一点 dx/dt = 0，那么分母为零，切线可能是竖直的。如果 dx/dt 和 dy/dt 同时为零，则该点称为奇异点，此公式无法唯一确定梯度。
In practice, the method is straightforward: differentiate x with respect to t, differentiate y with respect to t, then divide the second result by the first.
在实际操作中，方法很简单：先
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