📚 Parametric Differentiation | 参数方程求导
variable, often time t or an angle θ. These are parametric equations. To analyse the gradient, tangents, and the shape of such curves, we differentiate parametrically. This revision note covers the essential techniques, second derivatives, and common exam-style applications. Mastery of parametric differentiation also builds a solid foundation for later topics such as polar coordinates and vector calculus.
在许多 A-Level 题目中，曲线并不是通过 y = f(x) 这样的显函数给出，而是借助第三个变量（通常是时间 t 或角度 θ）来表达。这就叫参数方程。为了分析这类曲线的斜率、切线和形状，我们需要进行参数求导。本篇复习笔记涵盖了基本方法、二阶导数以及常见的考试应用题型。掌握参数求导还能为极坐标、向量微积分等后续内容打下扎实基础。
1. What Are Parametric Equations? | 什么是参数方程？
be written as x = cos θ, y = sin θ. This flexibility allows us to describe loops, self-intersections, and motion along a path more naturally than Cartesian equations alone.
用一个参数 t 同时决定 x 和 y。函数关系不再是由一个 y 关于 x 的等式给出，而是 x = f(t) 且 y = g(t)。例如单位圆可以写成 x = cos θ, y = sin θ。这种灵活性使得我们能够比单纯的直角坐标方程更自然地描述环路、自交图形以及沿轨道的运动。
Parametric curves frequently appear in mechanics (where t is time), in geometry (using an angle), and in calculus problems involving rates of change. Recognising the parameter's role is the first step before differentiation.
参数曲线经常出现在力学（t 表示时间）、几何（用角度表示）以及涉及变化率的微积分问题中。在进行求导前，首先要清楚参数扮演的角色。
2. The Chain Rule for Parametric Differentiation | 参数求导的链式法则
) ÷ (dx/dt), provided dx/dt ≠ 0. This formula is the core of all parametric differentiation. It reveals that the gradient is the ratio of the vertical rate of change to the horizontal rate of change.
既然 y 是 t 的函数，x 也是 t 的函数，那么我们就可以利用链式法则求出 dy/dx：dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。这个公式是所有参数求导的核心。它表明曲线斜率等于纵向变化率与横向变化率的比值。
dy/dx = (dy/dt) / (dx/dt)
Many students mistakenly flip the ratio. Remember: 'y over t' divided by 'x over t' gives 'y over x'. Practice deriving this from the chain rule: dy/dt = (dy/dx) · (dx/dt).
很多学生容易把比值颠倒。记住：“y对t”除以“x对t”得到“y对x”。多练习从链式法则推导这个关系：dy/dt = (dy/dx) · (dx/dt)。
3. Step-by-Step Differentiation of Parametric Curves | 参数曲线求导步骤
n form the quotient. Finally, if the question asks for the gradient at a specific point, find the corresponding t-value and substitute. It is crucial to keep all derivatives in terms of t until the last step to avoid messy algebra.
首先对 x = f(t) 求导得到 dx/dt，再对 y = g(t) 求导得到 dy/dt。然后构造商式。最后，如果题目要求某点的斜率，就找出对应的 t 值并代入。务必将对 t 的导数保持到最后一步再化简，以免代数运算混乱。
- Step 1: Compute dx/dt and dy/dt.
- Step 2: Write dy/dx = (dy/dt) / (dx/dt).
- Step 3: Simplify the expression, if possible.
- Step 4: Find t from the given x and y coordinates.
- Step 5: Evaluate dy/dx at that t.
- 步骤1：计算 dx/dt 和 dy/dt。
- 步骤2：写出 dy/dx = (dy/dt) / (dx/dt)。
- 步骤3：尽可能化简表达式。
- 步骤4：由给定的 x, y 坐标求出对应的 t。
- 步骤5：在该 t 值处计算 dy/dx。
4. Worked Example: A Basic Parametric Curve | 实例演练：基本参数曲线
dx = (3t² − 6) / (2t − 2). At t = 2, dy/dx = (3·4 − 6) / (4 − 2) = 6/2 = 3. The gradient is 3.
考虑 x = t² − 2t, y = t³ − 6t。求用 t 表示的 dy/dx，并确定 t = 2 时的梯度。首先，dx/dt = 2t − 2，dy/dt = 3t² − 6。于是 dy/dx = (3t² − 6) / (2t − 2)。当 t = 2 时，dy/dx = (3·4 − 6) / (4 − 2) = 6/2 = 3。梯度为 3。
t point the curve has a vertical tangent or a cusp; this is an important detail for sketching parametric curves.
注意，当 dx/dt = 0，即 t = 1 时导数无定义，曲线在该处可能存在铅直切线或尖点；这是勾画参数曲线时的一个重要细节。
5. Finding Equations of Tangents and Normals | 求切线与法线方程
erpendicular, has gradient −1/m. Both can be expressed in Cartesian form by substituting x₁ and y₁ from the parametric equations. Always quote the final answer in the required form.
当求出某点 (x₁, y₁) 处的 dy/dx 后，切线方程为 y − y₁ = m (x − x₁)，其中 m = dy/dx。法线与切线垂直，其斜率为 −1/m。代入由参数方程给出的 x₁ 和 y₁，两种线都可以化为直角坐标方程。最终答案一定要写成题目要求的格式。
− 0), or y = 3x − 4. The normal has slope −1/3, giving y + 4 = −1/3 x, i.e., y = −x/3 − 4.
例如，在上面那条曲线中，t = 2 时 x = 0, y = 8 − 12 = −4。切线为 y + 4 = 3(x − 0)，即 y = 3x − 4。法线斜率为 −1/3，得 y + 4 = −x/3，即 y = −x/3 − 4。
6. Second Derivative d²y/dx² in Parametric Form | 参数形式下的二阶导数 d²y/dx²
dt] ÷ (dx/dt). You must first find dy/dx as a function of t, then differentiate that with respect to t, and finally divide by dx/dt. Never simply divide d²y/dt² by d²x/dt² — that is a common mistake.
二阶导数衡量弯曲程度和凹凸性。在参数方程中，d²y/dx² = d(dy/dx)/dx = [d(dy/dx)/dt] ÷ (dx/dt)。必须先求出用 t 表示的 dy/dx，再将其对 t 求导，最后除以 dx/dt。绝不能直接用 d²y/dt² 除以 d²x/dt² —— 这是常见错误。
d²y/dx² = [d/dt (dy/dx)] / (dx/dt)
This expression is essential for classifying stationary points and for determining the convexity of a parametric segment. It is frequently tested in EdExcel A-Level Pure Mathematics Paper 2.
这个表达式对于判定驻点的类型以及确定参数曲线段的凹凸性至关重要。在 EdExcel A-Level 纯数卷二中经常考查。
7. Worked Example Involving the Second Derivative | 涉及二阶导数的例题
hen d(dy/dx)/dt = (3/2)(1 + 1/t²). Hence d²y/dx² = [(3/2)(1 + 1/t²)] / (2t) = (3/(4t))(1 + 1/t²) = 3(t² + 1)/(4t³). At t = 0 this is undefined, which signals a cusp or vertical tangent.
设 x = t², y = t³ − 3t。求用 t 表示的 d²y/dx²，并确定 t = 0 时的值。首先，dx/dt = 2t，dy/dt = 3t² − 3 → dy/dx = (3t² − 3)/(2t) = (3/2)(t − 1/t)。然后 d(dy/dx)/dt = (3/2)(1 + 1/t²)。因此 d²y/dx² = [(3/2)(1 + 1/t²)] / (2t) = (3/(4t))(1 + 1/t²) = 3(t² + 1)/(4t³)。当 t = 0 时无定义，表明该处存在尖点或铅直切线。
This neat simplification illustrates that d²y/dx² can often be expressed compactly, making it easier to analyse concavity. Pay attention to algebraic manipulation, as exam questions reward a clear chain of derivatives.
这个简洁的化简说明二阶导数常常可以表达得很紧凑，从而更易分析凹凸性。要重视代数操作，考试评分注重清晰的求导链条。
8. Stationary Points and Their Nature for Parametric Curves | 参数曲线的驻点及其性质
o classify it, evaluate d²y/dx² at that point: if positive, the point is a local minimum; if negative, a local maximum. If the second derivative is zero or undefined, additional tests with surrounding t-values are needed.
当 dy/dx = 0，即 dy/dt = 0（且 dx/dt ≠ 0）时，曲线出现驻点。为了对驻点进行分类，在该点处计算 d²y/dx²：若为正，是局部极小点；若为负，是局部极大点。若二阶导数为零或无定义，则需用周围的 t 值进行额外检验。
Because the parametric context might hide the actual shape, it is safer to construct a small table showing the sign of dy/dx on either side of the stationary t-value. This gradient test works universally and aligns with Edexcel mark schemes.
由于参数背景可能掩盖了实际形状，更为稳妥的做法是制作一个小表格，显示驻点 t 值两侧 dy/dx 的符号。这种梯度检验法普适且与 Edexcel 的评分方案一致。
9. Converting Parametric Equations to Cartesian Form | 将参数方程转化为直角坐标形式
² = 1. This is an ellipse. Elimination helps to recognise the curve and can simplify finding asymptotes or intersections.
有时候消去参数很有用。例如，由 x = 2 cos θ, y = 3 sin θ，利用恒等式 cos² θ + sin² θ = 1 可得到 (x/2)² + (y/3)² = 1，这是一条椭圆。消元有助于识别曲线的类型，也能简化渐近线或交点等问题。
However, conversion is not always straightforward. When x and y involve polynomials in t, you might need to solve for t from one equation and substitute into the other. The domain of the Cartesian equation should reflect the range of the original parameter.
然而，消元并非总是直接了当。当 x 和 y 为 t 的多项式时，可能需要从一个方程中解出 t 并代入另一个。所得到的直角坐标方程的定义域应反映原始参数的范围。
10. Applications to Kinematics and Motion | 在运动学中的应用
(dx/dt)² + (dy/dt)²]. The gradient dy/dx gives the direction of motion. Moreover, d²x/dt² and d²y/dt² are the acceleration components, while d²y/dx² relates to the curvature of the path, not directly to acceleration.
当 t 表示时间时，dx/dt 和 dy/dt 分别是速度的水平分量和垂直分量。速率为 √[(dx/dt)² + (dy/dt)²]。斜率 dy/dx 给出了运动方向。此外，d²x/dt² 和 d²y/dt² 是加速度分量，而 d²y/dx² 与路径的曲率有关，并非直接表示加速度。
This distinction is a classic trap: acceleration along the curve is not the same as the second derivative of y with respect to x. Keep the two concepts separate by using vector notation when necessary.
这种区别是一个经典陷阱：沿曲线的加速度并不等同于 y 对 x 的二阶导数。必要时使用向量表示，将这两个概念区分开。
11. Common Mistakes and How to Avoid Them | 常见错误与避免方法
| Mistake / 错误 | Correction / 纠正 |
|---|---|
| Using dy/dx = (dx/dt)/(dy/dt) | Always dy/dt divided by dx/dt. |
| Forgetting dx/dt = 0 cases | Check where dx/dt = 0; dy/dx undefined there. |
| Dividing second derivatives directly | Use d(dy/dx)/dt ÷ dx/dt. |
| Confusing d²y/dx² with acceleration | Acceleration uses d²x/dt² and d²y/dt². |
Careless algebra when simplifying the quotient for dy/dx is another frequent source of error. Factorise and cancel common terms early to keep expressions manageable. Always double-check that your final derivative is expressed in terms of the correct variable (usually t or θ).
在化简 dy/dx 的商式时代数不仔细是另一个常见错误来源。尽早因式分解并约去公因子，使表达式保持简洁。务必反复检查最终导数是否用正确的变量（通常是 t 或 θ）表达。
12. Exam Strategy and Tips | 考试策略与提示
Edexcel exam questions often guide you step by step: part (a) asks for dy/dx, part (b) for the equation of tangent or normal, and part (c) for the second derivative or stationary point classification. Follow the leads and always show the quotient for dy/dx clearly. Many marks are awarded for method.
Edexcel 的考题通常按步骤引导：(a)部分求 dy/dx，(b)部分求切线或法线方程，(c)部分涉及二阶导数或驻点分类。顺着引导走，并始终清楚地展示 dy/dx 的商式形式。很多分数都是给在方法上的。
When a problem gives a point in Cartesian coordinates, immediately find the parameter value(s). Some curves pass through a point for multiple t-values — each yields a different branch. Also, sketch the curve if it helps you visualise gradients and concavity. In pure mathematics papers, parametric differentiation is often linked with integration, so be prepared for volume or area problems that require eliminating the parameter or integrating parametrically.
当题目给出一个直角坐标点时，马上找出参数的值。有些曲线会以多个 t 值经过同一点——每个 t 值对应一条不同的分支。另外，如果有助于直观理解斜率和凹凸性，可以画出曲线草图。在纯数试卷中，参数求导经常与积分结合，因此要准备好解一些需要消元或参数积分的体积、面积问题。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图