Differentiation: Rules, Techniques and Applications | 微分：法则、技巧与应用

📚 Differentiation: Rules, Techniques and Applications | 微分：法则、技巧与应用

In Edexcel A-Level Mathematics, differentiation is a central Pure topic that connects gradients, rates of change, curve sketching and optimisation. This revision guide explains the key rules, standard derivatives and exam techniques you need for both AS and A2 papers.

在 Edexcel A-Level 数学中，微分是纯数学的核心主题，连接斜率、变化率、曲线作图与最优化。本复习指南讲解 AS 与 A2 试卷所需的关键法则、标准导数与考试技巧。

1. The Idea of Differentiation | 微分的基本思想

Differentiation measures the instantaneous rate of change of a function. For a curve y = f(x), the derivative f'(x) or dy/dx gives the gradient of the tangent at any point x. It is built from the limit of average rates of change over smaller and smaller intervals.

微分衡量函数在某一点的瞬时变化率。对于曲线 y = f(x)，导数 f'(x) 或 dy/dx 给出任意点 x 处切线的斜率。这一概念源于将平均变化率的区间不断缩小并取极限。

dy/dx = lim_h→0 [f(x+h) − f(x)] / h

2. First Principles and Power Rule | 第一原理与幂函数法则

Edexcel exams may ask you to differentiate a simple function from first principles. For f(x) = x², expand f(x+h) − f(x) = 2xh + h², divide by h and let h → 0 to obtain 2x. Once the definition is understood, we usually apply the power rule.

Edexcel 考试可能要求从第一原理出发对简单函数求导。例如 f(x) = x²，展开 f(x+h) − f(x) = 2xh + h²，除以 h 后令 h → 0，得到 2x。理解定义后，通常使用幂函数法则。

If y = xⁿ, then dy/dx = nxⁿ⁻¹

Multiplying by a constant simply multiplies the derivative, so the derivative of kxⁿ is knxⁿ⁻¹.

常数倍只乘在导数上，因此 kxⁿ 的导数为 knxⁿ⁻¹。

3. Standard Derivatives for Exponentials, Logarithms and Trigonometry | 指数、对数与三角函数的导数

You must memorise the standard derivatives below. In calculus, trigonometric functions are always in radians unless stated otherwise.

必须熟记以下标准导数。微积分中，若无特殊说明，三角函数一律使用弧度制。

For linear inner functions, use the chain rule shortcut: d/dx[sin(ax+b)] = a cos(ax+b).

对于线性内层函数，可使用链式法则的快捷形式：d/dx[sin(ax+b)] = a cos(ax+b)。

4. Chain Rule | 链式法则

The chain rule differentiates composite functions y = f(g(x)). Let u = g(x), then dy/dx = dy/du × du/dx. For powers, the derivative of [f(x)]ⁿ is n[f(x)]

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