Parametric Differentiation | 参数方程求导

📚 Parametric Differentiation | 参数方程求导

In A-Level Edexcel Mathematics, parametric equations describe the coordinates of a curve using a third variable, often t. Being able to differentiate such equations is essential for finding gradients, equations of tangents and normals, and stationary points. This article unpacks the concept, the chain rule derivation, and works through both first and second derivatives, with plenty of examples and exam-style applications.

在 A-Level Edexcel 数学中，参数方程用第三个变量（通常为 t）来描述曲线上点的坐标。掌握参数方程求导对于求斜率、切线法线方程以及驻点至关重要。本文剖析参数求导的概念、链式法则推导，并通过大量例题和应试应用讲解一阶导数和二阶导数。

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

xample, x = t², y = 2t + 1. As t varies, the point (x, y) traces a curve in the plane. This representation elegantly handles curves that are not functions in the usual sense, including loops and overlapping paths.

与将 y 直接写成 x 的函数不同，参数方程将 x 和 y 都用一个独立参数（常为 t 或 θ）表示，例如 x = t², y = 2t + 1。当 t 变化时，点 (x, y) 在平面上描绘出一条曲线。这种表示能够优雅地处理非函数曲线，包括环状和自交路径。

y = b sin t (an ellipse). The key challenge is to find dy/dx without eliminating the parameter, a technique that relies on the chain rule.

在 Edexcel 考试大纲中，你会遇到从简单二次关系到三角路径（如 x = a cos t, y = b sin t 的椭圆）的参数形式。关键挑战在于不消去参数而求出 dy/dx，这一技巧依赖于链式法则。

2. The Derivative dy/dx in Parametric Form | 参数形式下的导数 dy/dx

If x and y are both differentiable functions of a parameter t, the gradient of the curve dy/dx can be expressed in terms of the derivatives with respect to t. The fundamental formula is:

若 x 和 y 都是参数 t 的可微函数，曲线的梯度 dy/dx 可用关于 t 的导数表示，基本公式为：

dy/dx = (dy/dt) ÷ (dx/dt),   provided dx/dt ≠ 0

/dt. Rearranging yields the quotient form. Note the condition dx/dt ≠ 0, which avoids division by zero and corresponds to points where the tangent is vertical or the curve is not smooth.

该关系式直接由链式法则得出：dy/dt = dy/dx · dx/dt，重新整理即得商式形式。注意条件 dx/dt ≠ 0，这避免了除以零，并对应切线竖直或曲线不光滑的点。

It is vital to remember that you differentiate each of x and y separately with respect to t before forming the quotient. Many candidates mistakenly try to differentiate y directly with respect to x without the intermediate step.

必须记住，你是分别对 t 求导 x 和 y，再构成商式。许多考生错误地试图跳过中间步骤直接对 x 求导 y。

3. The Chain Rule Derivation | 链式法则推导

(t). The rate at which y changes with x can be seen as the ratio of the rate of change of y with t to that of x with t. Formally, using differentials:

考虑由 x = f(t), y = g(t) 给出的曲线。y 随 x 的变化率可以看作 y 对 t 的变化率与 x 对 t 的变化率之比。形式上讲，利用微分：

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

ent dx are both proportional to dt. Their ratio, in the limit as dt→0, gives the gradient.

这个结果可以直观理解：若你通过改变 t 沿曲线移动，竖直位移 dy 与水平位移 dx 均与 dt 成正比。在 dt→0 的极限下，它们的比值就是梯度。

The chain rule is the backbone of all parametric differentiation and also underpins related rates of change.

更严格的做法是写出 dy/dx = (dy/dt) · (dt/dx)，而 dt/dx = 1/(dx/dt)，从而得出公式。链式法则是所有参数求导的基石，也支撑着相关变化率。

4. Step-by-Step Guide to Parametric Differentiation | 参数求导分步指南

When tackling a parametric differentiation problem, follow these steps methodically:

处理参数求导问题时，请有条理地遵循以下步骤：

Step 1: Identify the parametric equations x(t) and y(t).
Step 2: Differentiate x with respect to t to obtain dx/dt.
Step 3: Differentiate y with respect to t to obtain dy/dt.
Step 4: Write dy/dx = (dy/dt) / (dx/dt), stating the condition dx/dt ≠ 0.
Step 5: Simplify the resulting expression, often cancelling common factors.
Step 6: If required, evaluate at a specific parameter value to find the gradient of the tangent.

步骤1：识别参数方程 x(t) 和 y(t)。
步骤2：对 x 关于 t 求导，得到 dx/dt。
步骤3：对 y 关于 t 求导，得到 dy/dt。
步骤4：写出 dy/dx = (dy/dt) / (dx/dt)，并注明条件 dx/dt ≠ 0。
步骤5：化简表达式，通常会约去公因子。
步骤6：若需要，代入特定参数值求切线斜率。

r t ≠ 0. This streamlined process avoids error and makes your working clear for examiners.

例如，若 x = t³ 且 y = t²，则 dx/dt = 3t²，dy/dt = 2t，因此 dy/dx = 2t / (3t²) = 2/(3t)（t ≠ 0）。这一简洁流程可避免错误，并使解题过程对阅卷人清晰明了。

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5. Illustrative Example 1: Trigonometric Curves | 实例1：三角曲线

os θ, y = 2 sin θ, where 0 ≤ θ < 2π. Here the parameter is θ, but the method is identical.

典型的 Edexcel 风格题目涉及圆或椭圆的参数方程。取 x = 2 cos θ, y = 2 sin θ，其中 0 ≤ θ < 2π。此处参数为 θ，但方法完全相同。

ngle θ. You can also verify that at θ = π/4, dy/dx = -1, which matches the geometry of the circle.

求导：dx/dθ = -2 sin θ，dy/dθ = 2 cos θ。于是 dy/dx = (2 cos θ) / (-2 sin θ) = -cot θ。这确认了圆在任意角度 θ 处的切线斜率。你还可以验证，在 θ = π/4 时，dy/dx = -1，与圆的几何性质一致。

particularly useful for finding points where the tangent is horizontal (cos t = 0) or vertical (sin t = 0).

一个更精巧的例子是椭圆：x = 3 cos t, y = 2 sin t。此时 dx/dt = -3 sin t, dy/dt = 2 cos t，所以 dy/dx = (2 cos t)/(-3 sin t) = -(2/3) cot t。该表达式对于求切线水平（cos t = 0）或竖直（sin t = 0）的点尤为有用。

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6. Illustrative Example 2: Algebraic Parametric Equations | 实例2：代数参数方程

ause dx/dt = 1 never vanishes, the derivative is defined for all t. Eliminating t gives y = (x - 2)² - 1, and the direct derivative 2(x - 2) indeed matches 2t since t = x - 2.

参数曲线不一定涉及三角函数。考虑由 x = t + 2, y = t² - 1 给出的抛物线。则 dx/dt = 1，dy/dt = 2t，得出 dy/dx = 2t。由于 dx/dt = 1 始终不为零，导数对所有 t 均有定义。消去 t 得 y = (x - 2)² - 1，直接求导得 2(x - 2) 确实与 2t 一致，因为 t = x - 2。

parametrisations require careful differentiation, often using the quotient rule. Practice with these sharpens algebraic manipulation skills, which are heavily tested in the Pure Mathematics examination.

另一种常见的代数形式是 x = (1 + t²)/t, y = (1 - t²)/t。这类有理参数化需要细心求导，常需使用商的求导法则。练习此类题目可强化代数操作能力，这在纯数学考试中考查颇多。

+ 1)/(t² - 1). Always simplify to reveal the behaviour of the gradient near singularities.

例如，若 x = t + 1/t 且 y = t - 1/t，则 dx/dt = 1 - 1/t²，dy/dt = 1 + 1/t²，从而 dy/dx = (1 + 1/t²)/(1 - 1/t²) = (t² + 1)/(t² - 1)。请务必化简，以揭示梯度在奇点附近的行为。

7. Finding the Equation of a Tangent | 求切线方程

ations to get the coordinates (x₀, y₀), and evaluate dy/dx at t₀ to obtain the gradient m. Then apply y - y₀ = m(x - x₀).

一旦用 t 表示的 dy/dx 求出后，就可以确定对应于 t = t₀ 的给定点处的切线方程。将 t₀ 代入参数方程得到坐标 (x₀, y₀)，并计算 t₀ 处的 dy/dx 得到斜率 m。然后应用 y - y₀ = m(x - x₀)。

= 1/t, so m = 1/2. Hence the tangent equation is y - 4 = (1/2)(x - 4), or y = (1/2)x + 2.

考虑曲线 x = t², y = 2t。对于 t = 2 的点，有 x₀ = 4, y₀ = 4。dy/dx = (2)/(2t) = 1/t，因此 m = 1/2。从而切线方程为 y - 4 = (1/2)(x - 4)，即 y = (1/2)x + 2。

m x = constant. Identify these points separately and state that the gradient is undefined.

当参数值导致 dx/dt = 0 时须小心。此时切线竖直，方程形如 x = 常数。应单独识别这些点，并说明斜率无定义。

8. Determining Stationary Points | 确定驻点

parameter value(s) that satisfy dy/dt = 0, then substitute back to find the (x, y) coordinates.

驻点出现在 dy/dx = 0 且曲线光滑处。使用参数求导，令 dy/dt = 0 且 dx/dt ≠ 0。解出满足 dy/dt = 0 的参数值，再代回求出 (x, y) 坐标。

at t = 0: dx/dt = -3 ≠ 0, so a stationary point exists. The coordinates are x = 0, y = 0.

对于曲线 x = t³ - 3t, y = t²，求导得 dx/dt = 3t² - 3, dy/dt = 2t。令 dy/dt = 0 得 t = 0。检查 t = 0 时的 dx/dt：dx/dt = -3 ≠ 0，因此存在驻点。坐标为 x = 0, y = 0。

Further classification (maximum, minimum, or point of inflection) requires the second derivative or sign changes of dy/dx around the parameter value. This is covered in the next section.

进一步判断驻点类型（极大、极小或拐点）需要利用二阶导数或检查参数值附近 dy/dx 的符号变化。下一节将介绍。

9. Second Derivative d²y/dx² from Parametric Form | 参数形式的二阶导数 d²y/dx²

The second derivative measures the rate of change of the gradient and is crucial for determining concavity and classifying stationary points. The formula for d²y/dx² in parametric form is derived by differentiating dy/dx with respect to x, again using the chain rule:

二阶导数衡量梯度的变化率，对确定凹凸性和驻点分类至关重要。参数形式的 d²y/dx² 公式由对 dy/dx 关于 x 求导并再次利用链式法则得出：

d²y/dx² = (d(dy/dx)/dt) / (dx/dt)

In practice, you first find dy/dx as an expression in t. Then differentiate that expression with respect to t to obtain d(dy/dx)/dt. Finally, divide by dx/dt. Never differentiate dy/dx directly with respect to x.

实践中，首先求出用 t 表示的 dy/dx。然后将该表达式对 t 求导得到 d(dy/dx)/dt。最后除以 dx/dt。切勿直接对 x 求导 dy/dx。

= 3/(4t). This shows that for t > 0 the curve is concave upwards, and for t < 0 concave downwards.

例：x = t², y = t³。则 dy/dx = (3t²)/(2t) = (3/2)t。将 dy/dx 对 t 求导：d(dy/dx)/dt = 3/2。dx/dt = 2t，所以 d²y/dx² = (3/2)/(2t) = 3/(4t)。这表明当 t > 0 时曲线向上凹，t < 0 时向下凹。

imum; if d²y/dx² < 0, it is a maximum. If it equals zero, further investigation is needed.

利用二阶导数对驻点分类时，代入使 dy/dt = 0 的参数值。若 d²y/dx² > 0，该点为极小值点；若 d²y/dx² < 0，则为极大值点。若等于零，则需进一步检验。

10. Application: Motion in a Plane | 应用：平面运动

parameter t often represents time, with x(t) and y(t) giving the coordinates. The velocity vector components are dx/dt and dy/dt, and the speed is √((dx/dt)² + (dy/dt)²). The direction of motion is given by dy/dx.

参数方程不仅限于几何；它们自然描述了平面上质点随时间运动的位置。在力学中，参数 t 常表示时间，x(t) 和 y(t) 给出坐标。速度向量分量为 dx/dt 和 dy/dt，速率大小为 √((dx/dt)² + (dy/dt)²)。运动方向由 dy/dx 给出。

For instance, if a particle' its path is dy/dx = -cot t. At t = π/2, dy/dx = 0, so the particle is moving horizontally.

例如，若质点在 t 秒时的位置为 x = cos t, y = sin t，它沿单位圆运动。其速度分量为 dx/dt = -sin t, dy/dt = cos t，轨迹梯度为 dy/dx = -cot t。当 t = π/2 时，dy/dx = 0，故质点朝水平方向运动。

Understanding the link between parametric differentiation and kinematics prepares you for problems that blend pure mathematics with applied contexts, a feature of modern Edexcel papers.

理解参数求导与运动学的联系，有助于你应对纯数学与应用情境结合的问题，这是当前 Edexcel 试卷的一大特色。

11. Common Pitfalls and Exam Tips | 常见错误与考试提示

Many students lose marks on parametric differentiation questions by making avoidable errors. Here are the most frequent pitfalls and how to dodge them:

许多学生在参数求导题目上因可避免的错误而失分。以下是最常见的陷阱及规避方法：

• Forgetting the condition dx/dt ≠ 0: Always state this when writing the formula for dy/dx. Examiners look for it.

  忘记条件 dx/dt ≠ 0：写下 dy/dx 公式时务必声明此条件。阅卷人会留意。

• Mishandling the second derivative: Never treat d²y/dx² as (d²y/dt²) / (d²x/dt²). Use the correct chain rule cascade.

  错误处理二阶导数：切勿将 d²y/dx² 当作 (d²y/dt²) / (d²x/dt²)。请使用正确的链式法则递推。

• Not simplifying dy/dx: Simplifying can reveal cancellations and avoid making later steps more complicated.

  未化简 dy/dx：化简可揭示可约项，避免后续步骤复杂化。

• Ignoring vertical tangents: and dy/dt ≠ 0, the tangent is vertical. State its equation rather than forcing a gradient.

  忽略竖直切线：当 dx/dt = 0 且 dy/dt ≠ 0 时，切线竖直。应写出其方程，而非强行给出斜率。

• Confusing the parameter with the variable: Keep track of whether you are differentiating with respect to t or x. A clear layout prevents this.

  混淆参数与变量：始终明确你是对 t 还是对 x 求导。清晰的书写布局可防止此类混淆。

Practising a variety of parametric forms—polynomial, trigonometric, and rational—will build confidence. Always verify your derivative by using a test point where possible, or by eliminating the parameter and comparing.

练习多样的参数形式——多项式、三角和有理式——将建立信心。尽可能通过测试点验证你的导数，或消去参数进行比较。

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