📚 Differentiation for Edexcel A-Level Maths | Edexcel A-Level 数学微分精讲
Differentiation is one of the most important topics in Edexcel A-Level Mathematics. It underpins many questions in Pure Mathematics and also appears in Mechanics, where you use derivatives to describe velocity and acceleration. This revision article covers the core differentiation techniques required by the Edexcel specification, including the chain rule, product rule, quotient rule, parametric differentiation, implicit differentiation, second derivatives, and key applications. Read through each section carefully and use the paired Chinese explanations to strengthen your understanding.
微分是 Edexcel A-Level 数学中最重要的主题之一。它是纯数学许多问题的基础，也出现在力学中，用导数描述速度和加速度。本篇复习文章涵盖 Edexcel 考试大纲要求的核心微分方法，包括链式法则、乘积法则、商法则、参数方程求导、隐函数求导、二阶导数及主要应用。请仔细阅读每一部分，并利用配对的中文解释加深理解。
1. The Derivative and Gradient | 导数与斜率
The derivative of a function f(x) is written as f'(x) or dy/dx. Geometrically, it gives the gradient of the tangent to the curve y = f(x) at any point x. The formal definition is based on the limit f'(x) = limₕ→₀ [f(x+h) – f(x)] / h, but at A-Level you mainly need to use standard results and rules.
函数 f(x) 的导数记为 f'(x) 或 dy/dx。从几何上看，它表示曲线 y = f(x) 在任意点 x 处切线的斜率。其正式定义基于极限 f'(x) = limₕ→₀ [f(x+h) – f(x)] / h，但在 A-Level 阶段你主要需要掌握标准结果和求导法则。
For a power function, the basic rule is:
对于幂函数，基本法则是：
d/dx (xⁿ) = nxⁿ⁻¹
This works for any real constant n, including negative and fractional powers. For example, d/dx(x⁻²) = -2x⁻³ and d/dx(√x) = d/dx(x¹ᐟ²) = ½x⁻¹ᐟ².
该法则适用于任意实数常数 n，包括负指数和分数指数。例如，d/dx(x⁻²) = -2x⁻³，且 d/dx(√x) = d/dx(x¹ᐟ²) = ½x⁻¹ᐟ²。
2. Basic Differentiation Rules | 基本微分法则
You should be able to differentiate sums, differences, and constant multiples directly. The derivative of a constant is zero, and the derivative of a constant multiple k·f(x) is k·f'(x).
你应该能够直接对和、差以及常数倍进行求导。常数的导数为零，常数倍 k·f(x) 的导数为 k·f'(x)。
Common results include:
常见结果包括：
| Function 函数 | Derivative 导数 |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| aˣ | aˣ ln a |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
Remember that for trigonometric functions, the variable must be measured in radians when you differentiate. This is especially important in questions involving gradients or numerical calculations.
请记住，对于三角函数，求导时变量必须以弧度为单位。这在涉及斜率或数值计算的问题中尤其重要。
3. The Chain Rule | 链式法则
The chain rule is used to differentiate composite functions, where one function is inside another. If y = f(u) and u = g(x), then:
链式法则用于求复合函数的导数，即一个函数在另一个函数内部。如果 y = f(u) 且 u = g(x)，则：
dy/dx = (dy/du) × (du/dx)
In practice, you differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function. For example, if y = (3x² + 2)⁵, let u = 3x² + 2, then dy/du = 5u⁴ and du/dx = 6x, so dy/dx = 30x(3x² + 2)⁴.
实际操作时，先对外层函数关于内层函数求导，再乘以内层函数的导数。例如，若 y = (3x² + 2)⁵，设 u = 3x² + 2，则 dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 30x(3x² + 2)⁴。
The chain rule can be applied repeatedly for nested functions. Edexcel often tests this with logs, exponentials, and trigonometric functions, such as y = e^(sin x) or y = ln(cos x).
链式法则可以反复应用于多层嵌套函数。Edexcel 经常将它与对数函数、指数函数和三角函数结合考查，例如 y = e^(sin x) 或 y = ln(cos x)。
4. The Product Rule | 乘积法则
When a function is the product of two differentiable functions u(x) and v(x), use the product rule:
当一个函数是两个可导函数 u(x) 与 v(x) 的乘积时，使用乘积法则：
d/dx (uv) = u dv/dx + v du/dx
For example, if y = x² sin x, then u = x² and v = sin x. Hence du/dx = 2x and dv/dx = cos x. The derivative is dy/dx = x² cos x + 2x sin x.
例如，若 y = x² sin x，则 u = x²，v = sin x。因此 du/dx = 2x，dv/dx = cos x。其导数为 dy/dx = x² cos x + 2x sin x。
You may need to factorise the result after applying the product rule. This is useful when finding stationary points or simplifying a derivative before a second differentiation.
在使用乘积法则后，你可能需要对结果进行因式分解。这在求驻点或在二次求导前化简导数时非常有用。
5. The Quotient Rule | 商法则
For a function expressed as a quotient y = u/v, where u and v are functions of x, the quotient rule states:
对于表示为商 y = u/v 的函数，其中 u 和 v 是关于 x 的函数，商法则为：
dy/dx = (v du/dx – u dv/dx) / v²
The order in the numerator is important: it is v times the derivative of u minus u times the derivative of v. A common mistake is reversing this order or forgetting to divide by v².
分子的顺序很重要：它是 v 乘以 u 的导数，再减去 u 乘以 v 的导数。常见的错误是颠倒顺序或忘记除以 v²。
For example, if y = x / (x² + 1), then du/dx = 1, dv/dx = 2x, so dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)².
例如，若 y = x / (x² + 1)，则 du/dx = 1，dv/dx = 2x，因此 dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)²。
Edexcel questions often combine the quotient rule with the chain rule, especially when the numerator or denominator contains a composite function.
Edexcel 考题经常将商法则与链式法则结合，尤其是当分子或分母包含复合函数时。
6. Parametric Differentiation | 参数方程求导
When a curve is defined parametrically by x = f(t) and y = g(t), you find dy/dx using the formula:
当曲线由参数方程 x = f(t) 和 y = g(t) 表示时，求 dy/dx 使用公式：
dy/dx = (dy/dt) / (dx/dt)
This is simply the chain rule in disguise: dy/dx = dy/dt × dt/dx. It works provided dx/dt is not zero at the point considered.
这实际上就是链式法则的变形：dy/dx = dy/dt × dt/dx。只要在所考虑的点处 dx/dt 不为零，该公式就成立。
For example, if x = t² + 1 and y = 2t³ – t, then dx/dt = 2t and dy/dt = 6t² – 1, so dy/dx = (6t² – 1) / 2t.
例如，若 x = t² + 1 且 y = 2t³ – t，则 dx/dt = 2t，dy/dt = 6t² – 1，因此 dy/dx = (6t² – 1) / 2t。
Edexcel usually asks you to find or use dy/dx for a parametric curve, and sometimes asks for the equation of the tangent or normal at a particular parameter value.
Edexcel 通常要求你求出或使用参数曲线的 dy/dx，有时还要求在某特定参数值处的切线或法线方程。
7. Implicit Differentiation | 隐函数求导
When an equation defines y implicitly as a function of x, such as x² + y² = 25, differentiate both sides with respect to x. Treat y as a function of x and use the chain rule for terms involving y.
当方程隐式地将 y 定义为 x 的函数时，例如 x² + y² = 25，对等式两边关于 x 求导。把 y 当作 x 的函数，并对含 y 的项使用链式法则。
For example, differentiating x² + y² = 25 gives 2x + 2y(dy/dx) = 0, so dy/dx = -x/y.
例如，对 x² + y² = 25 求导得到 2x + 2y(dy/dx) = 0，因此 dy/dx = -x/y。
For products involving x and y, use the product rule. For instance, in x²y³ = 1, differentiate to get 2xy³ + 3x²y²(dy/dx) = 0, then rearrange for dy/dx.
对于含有 x 和 y 的乘积，使用乘积法则。例如，在 x²y³ = 1 中，求导得到 2xy³ + 3x²y²(dy/dx) = 0，然后重新整理求出 dy/dx。
This technique is particularly useful for curves that cannot easily be expressed in the form y = f(x), and it is a regular feature of Edexcel Pure Mathematics papers.
这一方法对于难以写成 y = f(x) 形式的曲线特别有用，是 Edexcel 纯数学试卷中的常见题型。
8. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative, written as d²y/dx² or f”(x), is obtained by differentiating dy/dx again. It measures the rate of change of the gradient and is used to determine concavity.
二阶导数写作 d²y/dx² 或 f”(x)，是对 dy/dx 再次求导得到的。它衡量斜率的变化率，用于判断曲线的凹凸性。
If f”(x) > 0 on an interval, the curve is convex (or concave upward); if f”(x) < 0, the curve is concave (or concave downward).
如果在某区间内 f”(x) > 0，则该曲线是凸的（或向上凹）；如果 f”(x) < 0，则曲线是凹的（或向下凹）。
At a stationary point where f'(x) = 0, the second derivative test helps classify it. If f”(x) > 0, the point is a local minimum; if f”(x) < 0, the point is a local maximum; if f”(x) = 0, the test is inconclusive and you should consider the sign of f'(x) either side.
在驻点处 f'(x) = 0，二阶导数检验有助于判断该点的性质。若 f”(x) > 0，则该点是局部极小点；若 f”(x) < 0，则该点是局部极大点；若 f”(x) = 0，则该检验无法判断，你需要考察驻点两侧 f'(x) 的符号。
9. Applications: Tangents and Normals | 应用：切线与法线
Once you have dy/dx, you can find the equation of a tangent or normal to a curve at a given point. The gradient of the tangent at x = a is m = dy/dx evaluated at that point.
一旦求出 dy/dx，就可以求曲线在给定点处的切线或法线方程。x = a 处切线的斜率 m 等于 dy/dx 在该点处的值。
The equation of the tangent at (x₁, y₁) is y – y₁ = m(x – x₁). The normal is perpendicular to the tangent, so its gradient is -1/m, provided m ≠ 0.
在点 (x₁, y₁) 处的切线方程为 y – y₁ = m(x – x₁)。法线与切线垂直，因此法线斜率为 -1/m（前提是 m ≠ 0）。
Be careful with sign errors when finding the normal gradient. Also remember to express the final equation in the required form, such as ax + by + c = 0 or y = mx + c.
求法线斜率时要注意符号错误。同时记住最后将方程写成题目要求的形式，例如 ax + by + c = 0 或 y = mx + c。
10. Applications: Kinematics and Optimisation | 应用：运动学与优化
In Mechanics, if displacement s is given as a function of time t, then velocity v is ds/dt and acceleration a is dv/dt, which is also d²s/dt². These relationships are essential for motion in a straight line.
在力学中，若位移 s 表示为时间 t 的函数，则速度 v 为 ds/dt，加速度 a 为 dv/dt，也就是 d²s/dt²。这些关系是直线运动问题的基础。
For example, if s = 3t³ – 2t² + 5, then v = 9t² – 4t and a = 18t – 4. You can then substitute a given time to find velocity or acceleration at that instant.
例如，若 s = 3t³ – 2t² + 5，则 v = 9t² – 4t，a = 18t – 4。然后你可以代入给定的时间，求出该时刻的速度或加速度。
Differentiation is also used in optimisation problems, where you model a quantity, find its derivative, set it to zero, and determine maximum or minimum values. Always check that the critical point gives the required optimum, especially using the second derivative or sign analysis.
微分还用于优化问题：先建立某个量的模型，求其导数，令导数为零，并确定最大值或最小值。务必验证临界点确实给出所需的极值，尤其是使用二阶导数或符号分析进行判断。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
One common mistake is differentiating a product or quotient term by term, ignoring the product and quotient rules. Always scan the expression first and decide which rule applies.
一个常见的错误是逐项微分乘积或商，忽略了乘积法则和商法则。务必先观察表达式结构，再决定使用哪一条法则。
Another frequent error is using degrees instead of radians for trigonometric differentiation. In calculus, all angles are in radians unless the question explicitly states otherwise.
另一个常见错误是在三角函数的微分中使用角度制而不是弧度制。在微积分中，除非题目明确说明，否则所有角度均以弧度为单位。
When applying the quotient rule, write the numerator in brackets before simplifying. This reduces sign errors and helps you factorise correctly.
应用商法则时，在化简前先把分子写在括号内。这样可以减少符号错误，并帮助你正确分解因式。
In implicit differentiation, remember that every time you differentiate a term involving y, you must multiply by dy/dx. Missing this step is one of the most penalised errors.
在隐函数求导中，请记住每次对含有 y 的项求导时都必须乘以 dy/dx。漏掉这一步是最常被扣分的错误之一。
12. Worked Practice Examples | 典型练习示例
Example 1: Differentiate y = e^(2x) cos 3x. Here use the product rule with u = e^(2x) and v = cos 3x. Then du/dx = 2e^(2x) and dv/dx = -3 sin 3x. So dy/dx = e^(2x)(-3 sin 3x) + cos 3x(2e^(2x)) = e^(2x)(2 cos 3x – 3 sin 3x).
示例 1：对 y = e^(2x) cos 3x 求导。这里使用乘积法则，设 u = e^(2x)，v = cos 3x。则 du/dx = 2e^(2x)，dv/dx = -3 sin 3x。因此 dy/dx = e^(2x)(-3 sin 3x) + cos 3x(2e^(2x)) = e^(2x)(2 cos 3x – 3 sin 3x)。
Example 2: Find dy/dx for x³ + y³ = 6xy. Differentiate implicitly: 3x² + 3y²(dy/dx) = 6[y + x(dy/dx)]. Expand and rearrange to get dy/dx = (2y – x²) / (y² – 2x).
示例 2：求 x³ + y³ = 6xy 的 dy/dx。隐式求导：3x² + 3y²(dy/dx) = 6[y + x(dy/dx)]。展开并重新整理得到 dy/dx = (2y – x²) / (y² – 2x)。
Example 3: A curve is defined by x = t², y = 4t – 1. Find dy/dx in terms of t and the equation of the tangent at t = 1. Here dx/dt = 2t and dy/dt = 4, so dy/dx = 4 / (2t) = 2/t. At t = 1, the point is (1, 3) and the gradient is 2. The tangent is y – 3 = 2(x – 1), or y = 2x + 1.
示例 3：曲线由 x = t²，y = 4t – 1 定义。求用 t 表示的 dy/dx，并求 t = 1 处的切线方程。这里 dx/dt = 2t，dy/dt = 4，所以 dy/dx = 4 / (2t) = 2/t。当 t = 1 时，点为 (1, 3)，斜率为 2。切线方程为 y – 3 = 2(x – 1)，即 y = 2x + 1。
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