📚 Differentiation of Parametric Equations | 参数方程形式的函数求导
When a curve is expressed in parametric form, both x and y are given as functions of a third variable, usually t. We can still find the gradient dy/dx without eliminating t, using a simple quotient of derivatives. This is a core technique in A-Level and IB calculus, and it often appears in differentiation questions involving curves, tangents, normals and stationary points.
当曲线以参数方程形式给出时，x 与 y 都表示为第三个变量（通常为 t）的函数。我们不必消去参数 t，也能直接求出斜率 dy/dx；只需将两个导函数相除。这是 A-Level 和 IB 数学微积分中的核心技巧，也经常出现在涉及曲线、切线、法线与驻点的求导题中。
1. What Are Parametric Equations? | 什么是参数方程？
A curve can be written in Cartesian form as y = f(x), or implicitly as f(x, y) = 0. In parametric form, we use a third variable t to describe both coordinates at the same time.
曲线通常可以写成直角坐标形式 y = f(x)，或隐式形式 f(x, y) = 0。在参数形式中，我们引入第三个变量 t，同时描述两个坐标。
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Example: x = t², y = 2t. As t varies, the point (t², 2t) traces out a parabola.
例如：x = t²，y = 2t。当 t 变化时，点 (t², 2t) 描出一条抛物线。
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Example: x = r cos θ, y = r sin θ describes a circle when θ varies from 0 to 2π.
例如：x = r cos θ，y = r sin θ，当 θ 在 0 到 2π 间变化时，表示一个圆。
Parametric equations are especially useful in mechanics, because t can represent time. The position of a moving particle can be described by its x and y coordinates at each time t.
参数方程在力学中尤其有用，因为 t 可以表示时间。一个运动质点的位置，可以由它在每个时刻 t 的 x 坐标和 y 坐标来描述。
2. The Key Formula | 核心公式
Suppose a curve is given by x = f(t) and y = g(t). To find the gradient of the curve, we use the following formula.
设曲线由 x = f(t)，y = g(t) 给出。要求曲线的斜率，我们使用下面这个公式。
dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0
In words: differentiate y with respect to t, then divide by the derivative of x with respect to t.
换句话说：先对 t 求 y 的导数，再除以 x 对 t 的导数。
The condition dx/dt ≠ 0 is important. If dx/dt = 0, then a small change in t gives no change in x, so the tangent to the curve is vertical and dy/dx is not defined.
条件 dx/dt ≠ 0 很重要。如果 dx/dt = 0，那么 t 的微小变化不会引起 x 的变化，此时切线是竖直的，dy/dx 不存在。
3. Derivation: Why Does It Work? | 推导：为什么可行？
We can derive the formula using the chain rule. If y is considered as a function of x, and x is a function of t, then y is a composite function of t.
我们可以用链式法则来推导。如果把 y 看作 x 的函数，而 x 又是 t 的函数，那么 y 就是关于 t 的复合函数。
dy/dt = (dy/dx) × (dx/dt)
Dividing both sides by dx/dt gives the required formula.
两边同时除以 dx/dt，就得到所需公式。
Another way to see it is through small changes. Let Δt be a small change in t, producing small changes Δx and Δy.
我们也可以通过微小增量来理解。设 Δt 是 t 的微小改变量，它分别引起 Δx 和 Δy 的微小变化。
Δy/Δx = (Δy/Δt) / (Δx/Δt)
Taking the limit as Δt → 0 gives dy/dx = (dy/dt) / (dx/dt).
令 Δt → 0 取极限，即可得到 dy/dx = (dy/dt) / (dx/dt)。
4. Worked Example 1: Polynomial Parameter | 例 1：多项式参数
Find dy/dx for the curve x = t² + 1 and y = t³ + t. Evaluate the gradient at t = 2.
求曲线 x = t² + 1，y = t³ + t 的 dy/dx，并计算 t = 2 处的斜率。
Differentiate x and y with respect to t.
分别对 t 求 x 和 y 的导数。
dx/dt = 2t, dy/dt = 3t² + 1
Therefore:
因此：
dy/dx = (3t² + 1) / (2t)
At t = 2:
当 t = 2 时：
dy/dx = (12 + 1) / 4 = 13/4
The corresponding point on the curve is x = 5, y = 10. So the gradient at (5, 10) is 13/4.
对应的曲线点为 x = 5，y = 10。因此点 (5, 10) 处的斜率为 13/4。
Notice that if t = 0, then dx/dt = 0 and the gradient is undefined. This suggests a vertical tangent on the curve.
注意，如果 t = 0，则 dx/dt = 0，斜率不存在。这说明曲线上可能有一条竖直切线。
5. Tangents and Normals | 切线与法线
Once dy/dx is known, the tangent and normal lines can be found using the point corresponding to a given parameter value.
求出 dy/dx 之后，就可以利用给定参数值对应的点来求切线方程和法线方程。
For a point (x₀, y₀) with gradient m:
对于点 (x₀, y₀)，斜率为 m：
Tangent: y − y₀ = m(x − x₀)
Normal: y − y₀ = −(1/m)(x − x₀), m ≠ 0
Using the previous example at t = 2, we have x₀ = 5, y₀ = 10 and m = 13/4.
继续使用上一个例子，在 t = 2 处，x₀ = 5，y₀ = 10，m = 13/4。
The tangent equation is:
切线方程为：
y − 10 = (13/4)(x − 5)
This can be simplified to 4y = 13x − 25.
化简可得 4y = 13x − 25。
The normal has gradient −4/13, so its equation is:
法线的斜率为 −4/13，因此法线方程为：
y − 10 = −(4/13)(x − 5)
which simplifies to 13y + 4x = 150.
化简可得 13y + 4x = 150。
6. The Second Derivative | 二阶导数
The second derivative d²y/dx² measures how the gradient changes along the curve. It is used to determine concavity and points of inflection.
二阶导数 d²y/dx² 衡量的是曲线上斜率的变化率，常用于判断凹凸性和拐点。
To find d²y/dx², first write p = dy/dx as a function of t. Then:
要求 d²y/dx²，先把 p = dy/dx 写成 t 的函数，然后：
d²y/dx² = (dp/dt) / (dx/dt)
Equivalently:
等价地：
d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
A common mistake is to compute (d²y/dt²) / (d²x/dt²). This is wrong. The correct approach is to differentiate dy/dx with respect to t first, and then divide by dx/dt once more.
一个常见错误是直接去算 (d²y/dt²) / (d²x/dt²)。这是错误的。正确做法是先用 t 对 dy/dx 求导，然后再除以一个 dx/dt。
Continuing the earlier example, dy/dx = (3t² + 1)/(2t). We can rewrite this as:
继续前面的例子，dy/dx = (3t² + 1)/(2t)。我们可以将其改写为：
dy/dx = 3t/2 + 1/(2t)
Differentiate this with respect to t:
对 t 求导：
d/dt(dy/dx) = 3/2 − 1/(2t²)
Now divide by dx/dt = 2t:
再除以 dx/dt = 2t：
d²y/dx² = [3/2 − 1/(2t²)] / (2t) = 3/(4t) − 1/(4t³)
This gives the exact second derivative in terms of the parameter t.
这样就得到了以参数 t 表示的二阶导数。
7. Worked Example 2: Trigonometric Parameter | 例 2
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