Edexcel A-Level Maths: Master Differentiation and Its Applications | Edexcel A-Level 数学：掌握微分法及其应用

📚 Edexcel A-Level Maths: Master Differentiation and Its Applications | Edexcel A-Level 数学：掌握微分法及其应用

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It appears in questions on graphs, optimisation, rates of change, kinematics and proof. This guide covers the key rules, techniques and exam strategies you need to score full marks.

微分法是 Edexcel A-Level 数学中考查频率最高的主题之一。它出现在图像、最优化、变化率、运动学以及证明题中。本指南涵盖你需要掌握的核心法则、技巧和考试策略，帮助你拿到满分。

1. Differentiation from First Principles | 从第一性原理求导

The derivative of a function f(x) is defined as the limit of the average rate of change as h approaches zero. You may be asked to prove the derivative of x², √x or 1/x from first principles.

函数 f(x) 的导数定义为当 h 趋近于 0 时平均变化率的极限。考题可能会要求你从第一性原理证明 x²、√x 或 1/x 的导数。

f'(x) = lim (h→0) [f(x+h) – f(x)] / h

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h², so f'(x) = lim (h→0) (2xh + h²)/h = 2x.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²，因此 f'(x) = lim (h→0) (2xh + h²)/h = 2x。

2. Standard Derivatives and Basic Rules | 标准导数与基本法则

You must memorise the standard derivatives and the linearity rules: the derivative of a constant multiple and the derivative of a sum or difference.

你必须熟记标准导数以及线性法则：常数倍的导数以及和或差的导数。

Function f(x)	Derivative f'(x)
xⁿ	n xⁿ⁻¹
sin x	cos x
cos x	-sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x

Always write the rule you are using before applying it. For example, d/dx (5x³ – 2x + 7) = 15x² – 2.

在应用法则前要写出所使用的公式。例如，d/dx (5x³ – 2x + 7) = 15x² – 2。

3. The Chain Rule | 链式法则

The chain rule is used when one function is inside another, so y = f(g(x)). Let u = g(x), then y = f(u). The derivative is the product of the two derivatives.

当一个函数嵌套在另一个函数内部时使用链式法则，即 y = f(g(x))。设 u = g(x)，则 y = f(u)。导数是两个导数的乘积。

dy/dx = dy/du × du/dx

For example, if y = (3x² + 5)⁴, set u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 24x(3x² + 5)³.

例如，若 y = (3x² + 5)⁴，设 u = 3x² + 5。则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。

4. The Product and Quotient Rules | 乘积法则与商法则

When differentiating a product of two functions, use the product rule. When differentiating a fraction where both numerator and denominator contain x, use the quotient rule.

当对两个函数的乘积求导时，使用乘积法则。当分数的分子和分母都含有 x 时，使用商法则。

Product: d/dx (uv) = u dv/dx + v du/dx

Quotient: d/dx (u/v) = (v du/dx – u dv/dx) / v²

Example of the product rule: y = x² sin x gives dy/dx = x² cos x + 2x sin x.

乘积法则示例：y = x² sin x 得 dy/dx = x² cos x + 2x sin x。

Example of the quotient rule: y = x/(x² + 1) gives dy/dx = [(x² + 1)(1) – x(2x)]/(x² + 1)² = (1 – x²)/(x² + 1)².

商法则示例：y = x/(x² + 1) 得 dy/dx = [(x² + 1)(1) – x(2x)]/(x² + 1)² = (1 – x²)/(x² + 1)²。

5. Differentiation of Trigonometric Functions | 三角函数的微分

The derivatives of sin x, cos x and tan x must be known instantly. With the chain rule, they extend to sin(kx), cos(kx) and tan(kx).

sin x、cos x 和 tan x 的导数必须熟练掌握。结合链式法则，它们可以推广到 sin(kx)、cos(kx) 和 tan(kx)。

d/dx (sin kx) = k cos kx

d/dx (cos kx) = -k sin kx

d/dx (tan kx) = k sec² kx

In Edexcel exams, trigonometric differentiation is always carried out in radians unless the question explicitly states degrees.

在 Edexcel 考试中，三角函数求导总是在弧度制下进行，除非题目明确说明使用角度制。

6. Exponential and Logarithmic Differentiation | 指数与对数函数的微分

The exponential function eˣ is special because its derivative is eˣ. This makes growth and decay models easy to differentiate. The logarithm ln x differentiates to 1/x.

指数函数 eˣ 的特殊之处在于它的导数等于它本身。这使得增长和衰减模型易于求导。对数函数 ln x 的导数为 1/x。

d/dx (eᵏˣ) = k eᵏˣ

d/dx (ln x) = 1/x

For more complex expressions such as y = ln(2x + 1), use the chain rule: dy/dx = 2/(2x + 1).

对于 y = ln(2x + 1) 这类更复杂的表达式，使用链式法则：dy/dx = 2/(2x + 1)。

7. Parametric Differentiation | 参数方程求导

When a curve is defined parametrically by x = f(t) and y = g(t), the gradient is found by dividing the derivative of y with respect to t by the derivative of x with respect to t.

当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，导数等于 y 对 t 的导数除以 x 对 t 的导数。

dy/dx = (dy/dt) ÷ (dx/dt)

This formula is valid only where dx/dt ≠ 0. If dx/dt = 0, the tangent may be vertical.

该公式仅在 dx/dt ≠ 0 时成立。若 dx/dt = 0，切线可能是竖直的。

Example: x = t² + 1, y = 2t + 3. Then dx/dt = 2t, dy/dt = 2, so dy/dx = 1/t.

示例：x = t² + 1，y = 2t + 3。则 dx/dt = 2t，dy/dt = 2，因此 dy/dx = 1/t。

8. Stationary Points and Curve Sketching | 驻点与曲线草图

Stationary points occur where dy/dx = 0. These points can be local maxima, local minima or points of inflection. You need to classify them using the second derivative or a sign test.

驻点出现在 dy/dx = 0 处。这些点可能是局部极大值、局部极小值或拐点。你需要使用二阶导数或符号检验来判断驻点类型。

If f”(x) > 0 → local minimum

If f”(x) < 0 → local maximum

For example, y = x³ – 3x has dy/dx = 3x² – 3 = 3(x – 1)(x + 1), so stationary points are at x = 1 and x = -1. Since f”(x) = 6x, f”(1) = 6 > 0, so x = 1 is a minimum; f”(-1) = -6 < 0, so x = -1 is a maximum.

例如，y = x³ – 3x 的导数 dy/dx = 3x² – 3 = 3(x – 1)(x + 1)，因此驻点为 x = 1 和 x = -1。由于 f”(x) = 6x，f”(1) = 6 > 0，所以 x = 1 为极小值点；f”(-1) = -6 < 0，所以 x = -1 为极大值点。

9. Tangents and Normals | 切线与法线

At a point (x₁, y₁) on a curve, the tangent has gradient m = dy/dx evaluated at that point. Its equation is y – y₁ = m(x – x₁). The normal is perpendicular to the tangent, so its gradient is -1/m.

在曲线上一点 (x₁, y₁) 处，切线的斜率为该点处 dy/dx 的值 m。其方程为 y – y₁ = m(x – x₁)。法线垂直于切线，因此其斜率为 -1/m。

Tangent: y – y₁ = m(x – x₁)

Normal: y – y₁ = (-1/m)(x – x₁)

Example: for y = x² + 1 at x = 2, dy/dx = 2x so m = 4. The point is (2, 5). The tangent is y – 5 = 4(x – 2), and the normal is y – 5 = -¼(x – 2).

示例：对于 y = x² + 1 在 x = 2 处，dy/dx = 2x，因此 m = 4。点为 (2, 5)。切线为 y – 5 = 4(x – 2)，法线为 y – 5 = -¼(x – 2)。

10. Rates of Change and Connected Rates | 变化率与关联变化率

Connected rates of change problems require you to link different rates using the chain rule. Identify the quantity you want, then express it in terms of a variable whose rate of change is known.

关联变化率问题要求你使用链式法则将不同的变化率联系起来。先确定要求的变化率，然后将其表示为已知变化率的变量的函数。

dV/dt = dV/dr × dr/dt

For a sphere, V = 4/3 πr³, so dV/dr = 4πr². If dr/dt = 2 cm s⁻¹, then dV/dt = 4πr² × 2 = 8πr².

对于球体，V = 4/3 πr³，所以 dV/dr = 4πr²。若 dr/dt = 2 cm s⁻¹，则 dV/dt = 4πr² × 2 = 8πr²。

11. Second Derivative and Concavity | 二阶导数与凹凸性

The second derivative f”(x) describes the concavity of a curve. A positive second derivative means the curve is convex or valley-shaped; a negative second derivative means the curve is concave or hill-shaped.

二阶导数 f”(x) 描述曲线的凹凸性。二阶导数为正表示曲线向上凹或呈山谷形；二阶导数为负表示曲线向下凹或呈山峰形。

An inflection point occurs where f”(x) changes sign. At an inflection point, the curve changes from convex to concave or vice versa.

拐点出现在 f”(x) 改变符号的位置。在拐点处，曲线从凸变为凹，或从凹变为凸。

12. Exam Strategy for Edexcel Differentiation | Edexcel 微分题备考策略

Differentiation questions can be straightforward or multi-step. Always show each rule you use, simplify carefully, and check your answers by differentiation or substitution.

微分题可能是直接计算，也可能是多步骤问题。始终写出每一步使用的法则，仔细化简，并通过求导或代入检查答案。

Quote the product, quotient or chain rule before applying it.
应用乘积、商或链式法则之前先写出公式。
Use brackets to avoid sign errors in the quotient rule.
在商法则中使用括号，避免符号错误。
Check stationary point classification using the second derivative.
使用二阶导数检查驻点类型。
Always give exact answers unless a decimal approximation is requested.
除非题目要求近似值，否则给出精确答案。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Edexcel A-Level Maths: Master Differentiation and Its Applications | Edexcel A-Level 数学：掌握微分法及其应用