📚 A-Level Edexcel Maths: Parametric and Implicit Differentiation | 参数方程与隐函数求导
This revision guide covers parametric equations and implicit differentiation for Edexcel A-Level Mathematics. These topics appear regularly in Pure Mathematics 2 and Pure Mathematics 3, especially in questions that ask for gradients, tangents, normals, stationary points, and second derivatives. The examples below follow the style of exam-paper compilations such as PDF Joiner (4), page 235, where multiple skills are tested together.
本复习指南涵盖爱德思 A-Level 数学中的参数方程与隐函数求导。这些主题经常出现在 Pure Mathematics 2 和 Pure Mathematics 3 中，尤其是涉及斜率、切线、法线、驻点和二阶导数的题目。以下例题风格与试卷汇编（如 PDF Joiner (4) 第 235 页）一致，都是把多个技能结合在一道题中考查。
1. Syllabus Overview and Why It Matters | 考纲概览与重要性
Parametric differentiation and implicit differentiation are essential tools in the Edexcel A-Level Pure Mathematics specification. They are used to analyse curves that are not easily expressed as y = f(x). Candidates must be able to find dy/dx, d²y/dx², and use these to locate stationary points or write the equation of a tangent or normal.
参数方程求导和隐函数求导是爱德思 A-Level 纯数学考试大纲中的核心工具。它们用于分析那些不容易写成 y = f(x) 的曲线。考生必须能够求出 dy/dx、d²y/dx²，并用它们来求驻点或写出切线和法线的方程。
Common question types include: converting between parametric and Cartesian forms, finding gradients from parametric equations, differentiating implicit relations, and applying differentiation to coordinate geometry. Questions often combine parametric and implicit work with algebra, trigonometry, and logarithms.
常见题型包括：参数方程与笛卡尔形式的互化、从参数方程求斜率、对隐式关系求导，以及将求导应用于坐标几何。题目经常把参数方程和隐函数知识与代数、三角和对数等内容结合在一起。
2. Parametric Equations: The Basic Idea | 参数方程的基本思想
A parametric curve expresses x and y separately in terms of a third variable, usually t or θ. Instead of writing y = f(x), we write x = f(t) and y = g(t). This form is useful for describing motion, circles, ellipses, and many other curves.
参数曲线用第三个变量（通常是 t 或 θ）分别表示 x 和 y。我们不是写 y = f(x)，而是写 x = f(t) 和 y = g(t)。这种形式对于描述运动、圆、椭圆以及许多其他曲线非常有用。
For example, the parabola y² = 4ax can be written parametrically as x = at², y = 2at, where t is the parameter. In Edexcel questions, you may be given parametric equations and asked to eliminate t to find the Cartesian equation, or you may be asked to find a derivative directly.
例如，抛物线 y² = 4ax 可以用参数形式写成 x = at², y = 2at，其中 t 为参数。在爱德思考试题中，可能会给出参数方程，要求消去 t 求出笛卡尔方程，也可能要求直接求导数。
x = t² + 1, y = 2t − 3
Here each value of t gives one point on the curve. For example, when t = 2, x = 5 and y = 1, so the point (5, 1) lies on the curve.
这里每个 t 的值都给出曲线上的一个点。例如，当 t = 2 时，x = 5，y = 1，因此点 (5, 1) 在曲线上。
3. Differentiating Parametric Equations | 参数方程求导
To find the gradient of a parametric curve, we do not need to find y = f(x) first. We can use the chain rule in the form dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0. This is the key formula for parametric differentiation.
要求参数曲线的斜率，我们不必先求出 y = f(x)。我们可以使用链式法则，形式为 dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。这是参数方程求导的关键公式。
dy/dx = (dy/dt) ÷ (dx/dt) = (dy/dt) / (dx/dt)
Worked example: Given x = t², y = 2t, find dy/dx in terms of t. First compute dx/dt = 2t and dy/dt = 2. Then dy/dx = 2 ÷ 2t = 1/t.
例题：已知 x = t², y = 2t，求 dy/dx（用 t 表示）。先计算 dx/dt = 2t，dy/dt = 2。然后 dy/dx = 2 ÷ 2t = 1/t。
If the question asks for the gradient at a particular point, substitute the given t value into dy/dx. For example, at t = 3, dy/dx = 1/3.
如果题目要求某一点的斜率，就把给定的 t 值代入 dy/dx。例如，当 t = 3 时，dy/dx = 1/3。
4. Second Derivative for Parametric Curves | 参数曲线的二阶导数
The second derivative d²y/dx² measures the rate of change of the gradient and is used to determine concavity and nature of stationary points. For parametric curves, we must be careful: d²y/dx² is not simply the second derivative of y with respect to t. The correct formula is d²y/dx² = (d/dt[dy/dx]) ÷ (dx/dt).
二阶导数 d²y/dx² 衡量斜率的变化率，用于判断曲线的凹凸性和驻点的性质。对于参数曲线，我们必须注意：d²y/dx² 不是简单地对 t 求二阶导。正确的公式是 d²y/dx² = (d/dt[dy/dx]) ÷ (dx/dt)。
d²y/dx² = (d/dt[dy/dx]) / (dx/dt)
Using the previous example, x = t², y = 2t, we found dy/dx = 1/t. Then d/dt[1/t] = −1/t² and dx/dt = 2t. Therefore d²y/dx² = (−1/t²) ÷ (2t) = −1/(2t³).
继续使用前面的例子 x = t², y = 2t，我们得到 dy/dx = 1/t。然后 d/dt[1/t] = −1/t²，且 dx/dt = 2t。所以 d²y/dx² = (−1/t²) ÷ (2t) = −1/(2t³)。
5. Implicit Differentiation: When y Is Not Isolated | 隐函数求导：当 y 不显式出现时
Many curves are given by equations like x² + y² = 25 or x³ + 3xy + y³ = 6, where y is not written as a function of x. Implicit differentiation allows us to find dy/dx without solving for y. The method is to differentiate both sides of the equation with respect to x, treating y as a function of x and using the chain rule for any y-term.
许多曲线由 x² + y² = 25 或 x³ + 3xy + y³ = 6 这样的方程给出，其中 y 没有写成 x 的函数。隐函数求导使我们无需解出 y 就能求出 dy/dx。方法是对方程两边关于 x 求导，把 y 看成 x 的函数，并对每个 y 项使用链式法则。
For example, differentiate x² + y² = 25 with respect to x. The derivative of x² is 2x. The derivative of y² is 2y dy/dx, because y is a function of x. The right side differentiates to 0.
例如，对 x² + y² = 25 关于 x 求导。x² 的导数是 2x。y² 的导数是 2y dy/dx，因为 y 是 x 的函数。右边求导为 0。
2x + 2y(dy/dx) = 0
Solving gives dy/dx = −x/y. Notice that the answer still contains y. That is normal in implicit differentiation.
解得 dy/dx = −x/y。注意答案中仍然含有 y。这在隐函数求导中是完全正常的。
6. Standard Implicit Differentiation Worked Examples | 标准隐函数求导例题
Example: Find dy/dx for x³ + 3xy + y³ = 6. Differentiate term by term. The derivative of x³ is 3x². For 3xy, use the product rule: d/dx[3xy] = 3(y + x dy/dx). The derivative of y³ is 3y² dy/dx. The right side differentiates to 0.
例题：求 x³ + 3xy + y³ = 6 的 dy/dx。逐项求导。x³ 的导数是 3x²。对于 3xy，使用乘积法则：d/dx[3xy] = 3(y + x dy/dx)。y³ 的导数是 3y² dy/dx。右边求导为 0。
3x² + 3(y + x dy/dx) + 3y² dy/dx = 0
Collect all dy/dx terms on one side: 3x dy/dx + 3y² dy/dx = −3x² − 3y. Factor out dy/dx: dy/dx(3x + 3y²) = −3x² − 3y. Divide by 3: dy/dx(x + y²) = −(x² + y). Therefore dy/dx = −(x² + y)/(x + y²).
把所有 dy/dx 项移到一边：3x dy/dx + 3y² dy/dx = −3x² − 3y。提取公因式 dy/dx：dy/dx(3x + 3y²) = −3x² − 3y。两边除以 3：dy/dx(x + y²) = −(x² + y)。所以 dy/dx = −(x² + y)/(x + y²)。
This type of question often asks you to evaluate dy/dx at a given point. Substitute the coordinates into the final expression, not at an intermediate stage, to reduce errors.
这类题目经常要求在某一点求 dy/dx 的值。应把坐标代入最终表达式，而不要在中间步骤代入，以减少错误。
7. Tangents and Normals from Parametric and Implicit Curves | 参数与隐函数曲线的切线和法线
Once you have dy/dx, you can find the gradient of the tangent at a point. The tangent line passes through the point with gradient m = dy/dx. The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0.
一旦求出 dy/dx，就可以找到某一点处切线的斜率。切线经过该点，斜率为 m = dy/dx。法线垂直于切线，因此它的斜率为 −1/m，前提是 m ≠ 0。
Tangent: y − y₁ = m(x − x₁)
Normal: y − y₁ = (−1/m)(x − x₁)
For parametric curves, you must first find the point (x₁, y₁) by substituting the given t value into x(t) and y(t). Then calculate dy/dx at that t value. For implicit curves, the point is usually given directly.
对于参数曲线，必须先把给定的 t 值代入 x(t) 和 y(t) 求出点 (x₁, y₁)。然后在那个 t 值处计算 dy/dx。对于隐函数曲线，点通常是直接给出的。
8. Exam-Style Worked Problem | 考试型例题精讲
A curve is defined parametrically by x = t² + t and y = t³ − 3t. Find dy/dx in terms of t. Hence find the equation of the tangent to the curve at the point where t = 1.
曲线由参数方程 x = t² + t 和 y = t³ − 3t 定义。求 dy/dx（用 t 表示），并由此求出 t = 1 处的切线方程。
First, compute dx/dt = 2t + 1 and dy/dt = 3t² − 3. Then dy/dx = (3t² − 3)/(2t + 1).
首先，计算 dx/dt = 2t + 1，dy/dt = 3t² − 3。然后 dy/dx = (3t² − 3)/(2t + 1)。
At t = 1, dx/dt = 3 and dy/dt = 0, so dy/dx = 0. The point is x = 1² + 1 = 2 and y = 1³ − 3 = −2. The gradient of the tangent is 0, so the tangent is horizontal. Its equation is y = −2.
当 t = 1 时，dx/dt = 3，dy/dt = 0，所以 dy/dx = 0。点的坐标为 x = 1² + 1 = 2，y = 1³ − 3 = −2。切线斜率为 0，因此切线是水平的，其方程为 y = −2。
If the question asked for the normal at this point, the tangent is horizontal so the normal is vertical. A vertical line through (2, −2) has equation x = 2.
如果题目要求该点处的法线，由于切线是水平的，法线就是竖直的。过点 (2, −2) 的竖直直线方程为 x = 2。
9. Common Mistakes and How to Avoid Them | 常见错误与避坑指南
- Forgetting the chain rule for y-terms in implicit differentiation. — 在隐函数求导中忘记对 y 项使用链式法则。
- Dividing by dx/dt when it is zero. — 当 dx/dt 为零时仍然用它作除数。
- Sign errors when collecting dy/dx terms. — 在移项收集 dy/dx 项时出现符号错误。
- Misusing the second derivative formula for parametric curves. — 对参数曲线错误地使用二阶导数公式。
- Not finding the point coordinates from the parameter before writing the tangent. — 在写切线方程前没有先由参数求出点的坐标。
- Cancelling terms prematurely in implicit differentiation. — 在隐函数求导中过早约去项。
Always write down intermediate steps clearly. For implicit differentiation, put a bracket around each derivative of a y-term to avoid missing the dy/dx factor. For parametric differentiation, check that dx/dt is not zero before dividing.
始终把中间步骤写清楚。对于隐函数求导，每个 y 项的导数都要加括号，避免漏掉 dy/dx 因子。对于参数方程求导，在相除之前要检查 dx/dt 是否不为零。
10. Quick-Check Practice Questions | 快速自测题
Try these short questions to test your understanding. Answers are given below.
试一试下面这些简短的题目，检验你的掌握程度。答案见下方。
- Q1: Given x = 3t², y = 6t, find dy/dx. — 已知 x = 3t², y = 6t，求 dy/dx。
- Q2: For x² + xy + y² = 12, find dy/dx at the point (1, 2). — 对 x² + xy + y² = 12，求点 (1, 2) 处的 dy/dx。
- Q3: Given x = cos θ, y = sin θ, find d²y/dx². — 已知 x = cos θ, y = sin θ，求 d²y/dx²。
Answers: Q1: dy/dx = 1/t. Q2: dy/dx = −4/5. Q3: dy/dx = −cot θ, and d²y/dx² = −cosec³ θ.
答案： 第 1 题：dy/dx = 1/t。第 2 题：dy/dx = −4/5。第 3 题：dy/dx = −cot θ，d²y/dx² = −cosec³ θ。
11. Summary and Revision Checklist | 小结与复习清单
To master parametric and implicit differentiation, make sure you can do all of the following: find dy/dx from parametric equations; apply the second derivative formula correctly; differentiate implicit equations using the chain rule and product rule; find tangents and normals; and evaluate derivatives at given points or parameter values.
要掌握参数方程与隐函数求导，请确保你能完成以下所有内容：从参数方程求出 dy/dx；正确应用二阶导数公式；使用链式法则和乘积法则对隐式方程求导；求切线和法线；并在给定点或参数值处计算导数值。
| Formula / 公式 | Use / 用途 |
| dy/dx = (dy/dt)/(dx/dt) | Parametric gradient / 参数方程斜率 |
| d²y/dx² = (d/dt[dy/dx])/(dx/dt) | Parametric second derivative / 参数方程二阶导数 |
| 2x + 2y(dy/dx) = 0 | Implicit derivative example / 隐函数求导示例 |
Work through past-paper questions and mark schemes regularly. Focus on setting out your method clearly, because method marks are available even if the final answer is incorrect.
定期练习历年真题和评分方案。重点关注清晰的解题过程，因为即使最终答案错误，只要方法正确也能得到方法分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导