📚 Mastering Differentiation for Edexcel A Level Maths | 精通Edexcel A Level数学微分
Differentiation is one of the most heavily examined topics in the Edexcel A Level Mathematics specification. This article brings together the first-principles definition, the main derivative rules, applications to parametric and implicit functions, and the use of differentiation in optimisation problems. Each idea is paired in English and Chinese so you can revise efficiently.
微分是 Edexcel A Level 数学大纲中考查最频繁的主题之一。本文汇总第一性原理定义、主要求导法则、参数函数与隐函数的应用，以及微分在优化问题中的使用。每个要点均以英文和中文配对呈现，便于高效复习。
1. What Differentiation Measures | 微分衡量什么
Differentiation measures the instantaneous rate of change of a function. Geometrically, the derivative f ‘(x) gives the gradient of the tangent to the curve y = f(x) at the point (x, f(x)). If the gradient is positive, the function is increasing; if it is negative, the function is decreasing.
微分衡量函数的瞬时变化率。从几何上看，导数 f ‘(x) 表示曲线 y = f(x) 在点 (x, f(x)) 处切线的斜率。若斜率为正，函数递增；若斜率为负，函数递减。
The derivative can also be understood as a rate of change in applied contexts, such as velocity being the rate of change of displacement with respect to time. In pure mathematics, this idea connects directly to the gradient of a tangent.
在应用情境中，导数也可以理解为变化率，例如速度是位移关于时间的变化率。在纯数学中，这一概念与切线的斜率直接相关。
2. First Principles and the Gradient of a Curve | 从第一性原理与曲线的斜率
The derivative of f(x) is defined by the limit of the average rate of change as the interval h tends to zero. This is commonly known as differentiation from first principles.
函数 f(x) 的导数定义为当区间 h 趋于零时平均变化率的极限。这通常称为从第一性原理求导。
f ‘(x) = lim (h→0) [f(x+h) − f(x)] / h
To differentiate f(x) = x² from first principles, write f(x+h) − f(x) = (x+h)² − x² = 2xh + h². Divide by h to get 2x + h, then let h → 0, giving f ‘(x) = 2x.
若从第一性原理对 f(x) = x² 求导，先写出 f(x+h) − f(x) = (x+h)² − x² = 2xh + h²。除以 h 得到 2x + h，再令 h → 0，即得 f ‘(x) = 2x。
Examination questions often ask for this process using a simple polynomial. You must show the expansion, the cancellation of h, and the limiting step clearly to earn full marks.
考试题目常要求对简单多项式使用这一过程。你必须清楚展示展开、约去 h 以及取极限的步骤，才能获得满分。
3. Power Rule and Basic Polynomials | 幂法则与基本多项式
The power rule is the most frequently used rule in A Level differentiation. It states that to differentiate x to the power n, multiply by n and reduce the power by one.
幂法则是 A Level 微分中最常用的法则。它表示对 x 的 n 次方求导时，乘以 n，再将指数减一。
d/dx (xⁿ) = n xⁿ⁻¹
The rule works for any real value of n, including negative and fractional powers. For example, √x can be written as x to the power ½, so its derivative is ½ x to the power −½, which is the same as 1/(2√x).
该法则适用于任意实数 n，包括负指数和分数指数。例如，√x 可以写成 x 的 ½ 次方，因此其导数为 ½ x 的 −½ 次方，即 1/(2√x)。
For a constant a, the derivative of a xⁿ is a n xⁿ⁻¹, and the derivative of a constant function is zero. You should be able to differentiate expressions term by term.
对于常数 a，a xⁿ 的导数为 a n xⁿ⁻¹，常数函数的导数为零。你应能逐项对表达式求导。
4. Product, Quotient and Chain Rules | 乘法法则、除法法则与链式法则
When a function is the product of two functions u and v, use the product rule. The derivative is u prime v plus u v prime.
当一个函数是两个函数 u 和 v 的乘积时，使用乘法法则。其导数为 u 的导数乘以 v 再加上 u 乘以 v 的导数。
d/dx (u v) = u ′ v + u v ′
When a function is a quotient of two functions, use the quotient rule. The derivative is u prime v minus u v prime, all divided by v squared.
当一个函数是两个函数的商时，使用除法法则。其导数为 u 的导数乘以 v 减去 u 乘以 v 的导数，再除以 v 的平方。
d/dx (u/v) = (u ′ v − u v ′) / v²
The chain rule is used for composite functions. If y is a function of u and u is a function of x, then the derivative of y with respect to x is the product of dy/du and du/dx.
链式法则用于复合函数。若 y 是 u 的函数，u 是 x 的函数，则 y 关于 x 的导数为 dy/du 与 du/dx 的乘积。
dy/dx = dy/du × du/dx
For example, if y = (3x² + 1)⁵, set u = 3x² + 1, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x, giving dy/dx = 5(3x² + 1)⁴ × 6x.
例如，若 y = (3x² + 1)⁵，设 u = 3x² + 1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 5(3x² + 1)⁴ × 6x。
5. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The exponential function eˣ is unique because its derivative is itself. For a general exponential function aˣ, the derivative includes a factor of ln a.
指数函数 eˣ 的特殊之处在于其导数等于它本身。对于一般的指数函数 aˣ，导数中包含 ln a 的因子。
d/dx (eˣ) = eˣ and d/dx (aˣ) = aˣ ln a
The derivative of the natural logarithm ln x is 1/x. This is often combined with the chain rule, for example when differentiating ln(2x³ − 5).
自然对数 ln x 的导数为 1/x。这通常与链式法则结合使用，例如对 ln(2x³ − 5) 求导时。
d/dx (ln x) = 1/x
When differentiating ln u, where u is a function of x, the result is u ′ / u. This pattern is very common in Edexcel exam questions involving logarithmic differentiation.
当对 ln u 求导时，其中 u 是 x 的函数，结果为 u ′ / u。这种模式在涉及对数微分的 Edexcel 试题中非常常见。
6. Differentiating Trigonometric Functions | 三角函数的微分
Trigonometric differentiation requires the angle to be measured in radians. The three basic results for sin x, cos x and tan x must be memorised.
三角函数求导时角度必须使用弧度制。sin x、cos x 和 tan x 的三个基本结果必须牢记。
d/dx (sin x) = cos x d/dx (cos x) = −sin x d/dx (tan x) = sec² x
For reciprocal trigonometric functions, the standard results are d/dx (sec x) = sec x tan x, d/dx (cosec x) = −cosec x cot x, and d/dx (cot x) = −cosec² x.
对于倒数三角函数，标准结果为 d/dx (sec x) = sec x tan x，d/dx (cosec x) = −cosec x cot x，d/dx (cot x) = −cosec² x。
When the argument of a trigonometric function is not simply x, use the chain rule. For example, d/dx (sin 4x) = 4 cos 4x.
当三角函数的自变量不只是 x 时，使用链式法则。例如，d/dx (sin 4x) = 4 cos 4x。
7. Parametric Differentiation | 参数微分
When a curve is defined parametrically by x = f(t) and y = g(t), the gradient dy/dx is found by dividing dy/dt by dx/dt.
当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，斜率 dy/dx 可通过 dy/dt 除以 dx/dt 求得。
dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0
The second derivative d²y/dx² is not simply d²y/dt² divided by d²x/dt². Instead, differentiate dy/dx with respect to t, then divide by dx/dt.
二阶导数 d²y/dx² 不是简单地将 d²y/dt² 除以 d²x/dt²。正确的做法是先对 dy/dx 关于 t 求导，再除以 dx/dt。
d²y/dx² = [d/dt (dy/dx)] / (dx/dt)
For example, if x = t² and y = t³, then dx/dt = 2t and dy/dt = 3t², so dy/dx = (3t²)/(2t) = 3t/2.
例如，若 x = t²，y = t³，则 dx/dt = 2t，dy/dt = 3t²，因此 dy/dx = (3t²)/(2t) = 3t/2。
8. Implicit Differentiation | 隐函数微分
Implicit differentiation is used when y is not given as an explicit function of x. Differentiate both sides of the equation with respect to x, treating y as a function of x.
当 y 不是以显式函数形式给出时，使用隐函数微分。对等式两边关于 x 求导，并将 y 视为 x 的函数。
d/dx (yⁿ) = n yⁿ⁻¹ dy/dx
For example, for the circle x² + y² = 25, differentiating both sides gives 2x + 2y dy/dx = 0, so dy/dx = −x/y.
例如，对于圆 x² + y² = 25，两边求导得到 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。
You may also need to differentiate products such as x²y or xy² implicitly. In these cases, combine the product rule with the chain rule by including dy/dx whenever you differentiate a y term.
你还可能需要对 x²y 或 xy² 等乘积进行隐函数求导。此时应结合乘积法则和链式法则，只要对含 y 的项求导就应包含 dy/dx。
9. Stationary Points and Turning Points | 驻点与转折点
A stationary point occurs where the gradient of the curve is zero, that is where dy/dx = 0. Stationary points can be local maxima, local minima, or points of inflection.
驻点出现在曲线斜率为零的位置，即 dy/dx = 0 处。驻点可能是局部极大值、局部极小值或拐点。
To determine the nature of a stationary point, you can use the second derivative test. If d²y/dx² > 0 at the point, it is a minimum; if d²y/dx² < 0, it is a maximum.
要判断驻点的性质，可以使用二阶导数判别法。若该点处 d²y/dx² > 0，则为极小值；若 d²y/dx² < 0，则为极大值。
Alternatively, study the sign of dy/dx before and after the point. If the gradient changes from negative to positive, the point is a minimum; if it changes from positive to negative, it is a maximum.
另一种方法是研究该点前后 dy/dx 的符号变化。若斜率由负变正，该点为极小值；若由正变负，则为极大值。
10. Modelling and Optimisation Problems | 建模与优化问题
Differentiation is widely used in A Level modelling questions. A typical problem asks you to express a quantity such as area, volume, cost, or profit as a function of one variable, then find its maximum or minimum.
微分广泛用于 A Level 建模题中。典型题目要求你将面积、体积、成本或利润等量表示为单一变量的函数，然后求其最大值或最小值。
First, use the given constraints to eliminate extra variables. Then differentiate with respect to the remaining variable, set the derivative equal to zero, and solve for stationary points. Finally, justify whether each stationary point is a maximum or minimum using the second derivative or a sign test.
首先利用给定约束消去多余变量。然后对剩余变量求导，令导数等于零，解出驻点。最后利用二阶导数或符号检验判断每个驻点是极大值还是极小值。
In practical problems, you should also check the endpoints of the domain if the variable is restricted. The maximum or minimum may occur at an endpoint rather than at a stationary point.
在实际问题中，如果变量受到限制，还应检查区间的端点。最大值或最小值可能出现在端点而不是驻点。
11. Common Mistakes and Exam Tips | 常见错误与考试技巧
Avoid these frequent errors in differentiation questions. First, forgetting to multiply by the derivative of the inner function when using the chain rule. Second, using the quotient rule with the wrong sign in the numerator.
在微分题中要避免以下常见错误。第一，使用链式法则时忘记乘以内层函数的导数。第二，使用除法法则时分子中的符号出错。
- Write clear working for first-principles limits; do not skip the h → 0 step. | 第一性原理极限要写出清晰步骤，不要跳过 h → 0 的过程。
- Use radians for trigonometric differentiation, not degrees. | 三角函数求导要使用弧度制，不要使用角度制。
- For parametric second derivatives, differentiate dy/dx with respect to t first. | 参数方程的二阶导数要先对 t 求 dy/dx 的导数。
- In implicit differentiation, include dy/dx every time you differentiate a y term. | 隐函数求导中，每次对含 y 的项求导都要包含 dy/dx。
- Check stationary point nature with the second derivative or a gradient sign test. | 用二阶导数或斜率符号检验来判断驻点性质。
12. Quick Revision Checklist | 快速复习清单
Use this checklist to make sure you are ready for Edexcel differentiation questions. You should be able to differentiate polynomials, negative and fractional powers, exponentials, logarithms, and trigonometric functions.
使用这份清单确保你已准备好应对 Edexcel 微分题。你应能对多项式、负指数和分数指数、指数函数、对数函数及三角函数求导。
- Know the limit definition of derivative. | 掌握导数的极限定义。
- Apply product, quotient and chain rules accurately. | 准确运用乘积法则、商法则和链式法则。
- Differentiate parametric equations and find second derivatives. | 对参数方程求导并求二阶导数。
- Use implicit differentiation for equations involving both x and y. | 对含 x 和 y 的方程使用隐函数求导。
- Locate stationary points and determine their nature. | 找出驻点并判断其性质。
- Set up and solve optimisation problems from written models. | 根据文字模型建立并求解优化问题。
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