IB Maths: Derivatives of Parametric Functions | IB数学：参数式函数的导数

📚 IB Maths: Derivatives of Parametric Functions | IB数学：参数式函数的导数

Parametric equations are a powerful way to describe curves in the coordinate plane. Instead of writing y directly as a function of x, we write both x and y in terms of a third variable, usually called t. In this article we will develop the derivative rule for parametric functions, apply it to important examples, and identify common IB exam traps.

参数方程是描述坐标平面中曲线的一种强大方式。我们不直接把 y 写成 x 的函数，而是将 x 和 y 都用第三个变量（通常记为 t）表示。在本文中，我们将推导参数函数的求导法则，通过重要例题加以应用，并指出 IB 考试中常见的易错点。

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

For a curve described parametrically, we have a pair of equations x = f(t) and y = g(t). As t varies over an interval, the point (x, y) traces out a curve. The variable t often represents time, but it can also represent an angle or a distance along a curve.

对于用参数方式描述的曲线，我们有一对方程 x = f(t) 和 y = g(t)。当 t 在一个区间内变化时，点 (x, y) 会描绘出一条曲线。变量 t 通常表示时间，但也可以是角度或沿曲线行进的距离。

For example, x = cos t and y = sin t with 0 ≤ t ≤ 2π produce a unit circle. The parameter t removes the need to solve for y explicitly.

例如，x = cos t，y = sin t（0 ≤ t ≤ 2π）可以得到单位圆。参数 t 使我们在作图时无需显式解出 y。

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2. Why Differentiate Parametric Functions? | 为什么要对参数函数求导？

When asked for the gradient of a parametric curve, we need dy/dx. It tells us the slope of the tangent at any point. Notice that dy/dx is not obtained by eliminating t first, although elimination can be used for simple cases.

当题目要求参数曲线的斜率时，我们需要 dy/dx。它告诉我们曲线上任意一点处切线的斜率。注意，虽然消去 t 这种方法在简单情形下可行，但 dy/dx 并不是通过先消去 t 得到的。

Parametric differentiation is also useful in kinematics and related-rates problems, where x and y both depend on t.

参数求导在运动学和相关变化率问题中也很有用，因为在这些问题中 x 和 y 都依赖 t。

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3. The Core Formula | 核心公式

If x = f(t) and y = g(t), then the derivative of y with respect to x is:

若 x = f(t)，y = g(t)，则 y 对 x 的导数为：

dy/dx = (dy/dt) / (dx/dt), for dx/dt ≠ 0

Equivalently, using derivatives of f and g, we can write

等价地，使用 f 和 g 的导数，我们可以写成

dy/dx = g'(t) / f'(t), provided f'(t) ≠ 0

This formula is the single most important tool in this topic.

这个公式是本专题中最重要的工具。

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4. Derivation from the Chain Rule | 从链式法则推导

The formula follows directly from the chain rule. If y is a function of x and x is a function of t, then

该公式可以直接由链式法则推出。如果 y 是 x 的函数，而 x 是 t 的函数，那么

dy/dt = (dy/dx) × (dx/dt)

Dividing both sides by dx/dt gives the required formula, assuming dx/dt is not zero.

在 dx/dt 不为零的前提下，等式两边同时除以 dx/dt 即可得到所需公式。

This derivation explains why we must not simply differentiate y with respect to t and stop; we need the factor 1/(dx/dt).

这一推导解释了为什么不能只对 y 关于 t 求导就结束；我们还需要乘上因子 1/(dx/dt)。

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5. Worked Example 1: A Straight Line | 例题1：一条直线

Consider the parametric equations x = 2t + 1 and y = 4t − 3.

考虑参数方程 x = 2t + 1，y = 4t − 3。

We first differentiate with respect to t:

我们先对 t 求导：

dx/dt = 2, dy/dt = 4

Therefore:

因此：

dy/dx = 4/2 = 2

The slope is constant, as expected for a straight line. At t = 0, the point is (1, −3) and the tangent is the original line itself.

斜率是常数，这符合直线的情形。当 t = 0 时，

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