A-Level Edexcel Pure Maths: Mastering Differentiation Techniques | A-Level Edexcel 纯数学：掌握微分技巧

📚 A-Level Edexcel Pure Maths: Mastering Differentiation Techniques | A-Level Edexcel 纯数学：掌握微分技巧

Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. It underpins gradient calculations, rate-of-change problems, optimisation and curve sketching. This revision guide walks through the key rules and common exam pitfalls, with each rule paired with a concise Chinese explanation.

微分是 Edexcel A-Level 数学中考查频率最高的主题之一。它是梯度计算、变化率问题、优化和曲线草图的基础。本复习指南梳理关键法则与常见考试陷阱，每条法则均配有简明中文解释。

1. Differentiation from First Principles | 从第一性原理出发的微分

of the average rate of change as the interval shrinks to zero. The formal definition is f'(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h.

从第一性原理出发的微分通过令区间趋近于零，求曲线在某点的平均变化率的极限，从而得到梯度。正式定义为 f'(x) = lim (h → 0) [f(x + h) − f(x)] ÷ h。

ample, if f(x) = x², expanding f(x + h) = (x + h)² = x² + 2xh + h² and simplifying gives f'(x) = 2x. Edexcel often asks for this derivation in the Pure 1 paper.

例如，若 f(x) = x²，展开 f(x + h) = (x + h)² = x² + 2xh + h² 并化简可得 f'(x) = 2x。Edexcel 常在 Pure 1 试卷中要求写出此推导过程。

---

2. The Power Rule and Constant Multiple Rule | 幂法则与常数倍法则

The power rule states that if f(x) = xⁿ for any real constant n, then f'(x) = n xⁿ⁻¹. This rule is the fastest way to differentiate polynomial terms.

幂法则指出，若 f(x) = xⁿ（n 为任意实常数），则 f'(x) = n xⁿ⁻¹。该法则是微分多项式各项最快的方法。

You can also pull out a constant multiplier: if y = a f(x), then dy/dx = a f'(x). For instance, y = 5x³ gives dy/dx = 15x².

常数倍可以提出：若 y = a f(x)，则 dy/dx = a f'(x)。例如 y = 5x³ 时，dy/dx = 15x²。

f(x)	f'(x)
xⁿ	n xⁿ⁻¹
eᵏˣ	k eᵏˣ
ln x	1/x
sin x	cos x
cos x	−sin x
tan x	sec² x

---

3. The Chain Rule | 链式法则

The chain rule differentiates composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) × g'(x). In Leibniz notation, dy/dx = (dy/du) × (du/dx).

链式法则用于复合函数求导。若 y = f(g(x))，则 dy/dx = f'(g(x)) × g'(x)。用莱布尼茨记号表示为 dy/dx = (dy/du) × (du/dx)。

Let u = 3x² + 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x, giving dy/dx = 24x(3x² + 5)³.

例如 y = (3x² + 5)⁴。令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x，故 dy/dx = 24x(3x² + 5)³。

---

4. The Product Rule | 积法则

unctions multiplied together. If y = u v, where u and v are functions of x, then dy/dx = u'v + uv'.

积法则用于两个函数相乘时的求导。若 y = u v，其中 u 和 v 均为 x 的函数，则 dy/dx = u'v + uv'。

For y = x² eˣ, take u = x² and v = eˣ. Then u' = 2x and v' = eˣ, so dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x).

对于 y = x² eˣ，令 u = x²、v = eˣ。则 u' = 2x、v' = eˣ，因此 dy/dx = 2x eˣ + x² eˣ = x eˣ(2 + x)。

---

广告 / Sponsored

5. The Quotient Rule | 商法则

e quotient rule handles functions divided by another function. If y = u/v, then dy/dx = (u'v − uv') ÷ v². Remember the numerator is 'bottom times derivative of top minus top times derivative of bottom'.

商法则处理一个函数除以另一个函数的情况。若 y = u/v，则 dy/dx = (u'v − uv') ÷ v²。记住分子为“底乘顶的导数减顶乘底的导数”。

For y = x / (x + 1), u = x, v = x + 1, so u' = 1, v' = 1. Thus dy/dx = (1(x+1) − x(1)) ÷ (x+1)² = 1 ÷ (x+1)².

对于 y = x / (x + 1)，u = x、v = x + 1，则 u' = 1、v' = 1。因此 dy/dx = (1(x+1) − x(1)) ÷ (x+1)² = 1 ÷ (x+1)²。

---

6. Differentiating Exponentials and Logarithms | 指数函数与对数函数的微分

The derivative of eˣ is eˣ. More generally, if y = eᵏˣ, then dy/dx = k eᵏˣ. This is a direct application of the chain rule.

eˣ 的导数是 eˣ。更一般地，若 y = eᵏˣ，则 dy/dx = k eᵏˣ。这是链式法则的直接应用。

The derivative of ln x is 1/x for x > 0. For y = ln(f(x)), the chain rule gives dy/dx = f'(x) ÷ f(x).

ln x 的导数为 1/x（x > 0）。对于 y = ln(f(x))，由链式法则得 dy/dx = f'(x) ÷ f(x)。

Example: y = ln(3x² + 1). Here f(x) = 3x² + 1 and f'(x) = 6x, so dy/dx = 6x ÷ (3x² + 1).

例如 y = ln(3x² + 1)。这里 f(x) = 3x² + 1，f'(x) = 6x，故 dy/dx = 6x ÷ (3x² + 1)。

---

7. Differentiating Trigonometric Functions | 三角函数的微分

d/dx(cos x) = −sin x, and d/dx(tan x) = sec² x. These are in the Edexcel formula booklet.

标准三角函数的导数为：d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(tan x) = sec² x。这些在 Edexcel 公式手册中给出。

example, y = sin 2x gives dy/dx = 2 cos 2x; y = cos(3x + 1) gives dy/dx = −3 sin(3x + 1).

对于复合角，使用链式法则。例如 y = sin 2x 时 dy/dx = 2 cos 2x；y = cos(3x + 1) 时 dy/dx = −3 sin(3x + 1)。

---

8. Implicit Differentiation | 隐函数微分

spect to x, applying the chain rule to any y term: d/dx(y) = dy/dx, d/dx(y²) = 2y (dy/dx).

隐函数微分用于 y 未显式写成 x

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）