Edexcel A-Level Maths: Differentiation Techniques and Applications | 爱德思A-Level数学：微分技巧与应用

📚 Edexcel A-Level Maths: Differentiation Techniques and Applications | 爱德思A-Level数学：微分技巧与应用

Differentiation is a cornerstone of Edexcel A-Level Pure Mathematics. It appears in nearly every exam session, often in combination with integration, coordinate geometry, and modelling. A solid command of differentiation rules and their applications is essential for both Pure Paper 1 and Pure Paper 2.

微分是爱德思A-Level纯数学的核心内容，几乎每次考试都会出现，并常与积分、坐标几何和建模结合考查。扎实掌握微分法则及其应用对于纯数试卷1和试卷2至关重要。

1. Understanding the Derivative | 理解导数的含义

The derivative f'(x) measures the rate at which a function f(x) changes with respect to x. Geometrically, it gives the gradient of the tangent line to the curve y = f(x) at a chosen point.

导数 f'(x) 衡量函数 f(x) 随 x 变化的速率。从几何角度看，它表示曲线 y = f(x) 在所选点处切线的斜率。

In applied contexts, the derivative can represent velocity, acceleration, marginal cost, or any other rate of change. Edexcel examiners often expect you to interpret the derivative in a real-world situation rather than just compute it.

在应用情境中，导数可以表示速度、加速度、边际成本或其他变化率。爱德思考官通常希望你解释导数在实际问题中的意义，而不仅仅是计算它。

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

The formal definition of the derivative is a limit:

导数的正式定义是一个极限：

f'(x) = limₕ→₀ [f(x+h) − f(x)] / h

This definition often appears as a short question in Edexcel Pure Paper 1. You may be asked to prove the derivative of x² or a similar simple function from first principles.

这一定义经常在爱德思纯数试卷1中以简答题出现。你可能会被要求从第一性原理证明 x² 或类似简单函数的导数。

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h². Subtracting f(x) and dividing by h gives 2x + h. As h tends to 0, the limit is 2x.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。减去 f(x) 并除以 h 得到 2x + h。当 h 趋向于 0 时，极限为 2x。

3. The Power Rule and Basic Results | 幂函数法则与基本结果

The power rule is the most frequently used differentiation tool:

幂函数法则是使用频率最高的求导工具：

d/dx (xⁿ) = n xⁿ⁻¹

This rule works for any real value of n, including negative and fractional powers. You should rewrite terms such as 1/x² as x⁻² and √x as x¹ᐟ² before differentiating.

该法则适用于任何实数 n，包括负指数和分数指数。求导前应先将 1/x² 改写为 x⁻²，将 √x 改写为 x¹ᐟ²。

Basic derivatives that appear constantly include:

以下基本导数经常出现：

• d/dx (eˣ) = eˣ | d/dx (eˣ) = eˣ
• d/dx (ln x) = 1/x | d/dx (ln x) = 1/x
• d/dx (sin x) = cos x | d/dx (sin x) = cos x
• d/dx (cos x) = −sin x | d/dx (cos x) = −sin x
• d/dx (tan x) = sec² x | d/dx (tan x) = sec² x

Here the vertical bar separates English and Chinese equivalents for quick reference.

这里用竖线分隔英文和中文对应内容，便于快速查阅。

4. Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(u)

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