Edexcel A-Level Maths: Differentiation and Stationary Points | 爱德思 A-Level 数学：微分与驻点

📚 Edexcel A-Level Maths: Differentiation and Stationary Points | 爱德思 A-Level 数学：微分与驻点

In Edexcel A-Level Mathematics, differentiation is a central Pure topic that appears in Paper 1 and Paper 2. Questions often ask you to find gradients, tangents, normals, stationary points, and to classify maxima and minima. This revision guide focuses on the techniques and exam strategies needed for these questions.

在爱德思 A-Level 数学中，微分是纯数部分的核心内容，常出现在试卷一和试卷二中。题目通常要求计算梯度、切线、法线、驻点，并判断极大值和极小值。本复习指南重点讲解这些题型所需的方法和考试策略。

1. The derivative as a gradient function | 导数作为梯度函数

The derivative of a function y = f(x) measures the rate at which y changes with respect to x. Geometrically, it is the gradient of the tangent to the curve at any point. In Edexcel exams, the notations f'(x) and dy/dx are both used.

函数 y = f(x) 的导数度量 y 随 x 的变化率。从几何上看，它是曲线在任意一点处切线的斜率。在爱德思考试中，f'(x) 和 dy/dx 两种记号都会使用。

f'(x) = lim (h→0) [f(x+h) – f(x)] / h

This limit definition is not often asked directly, but understanding it helps with first principles questions and with interpreting gradient functions.

这个极限定义虽然不常直接考查，但理解它有助于解决第一原理题和解释梯度函数。

2. Standard derivatives and rules | 标准导数与法则

Edexcel expects you to recall the standard derivatives for powers, exponentials, logarithms and trigonometric functions. The power rule is used most often.

爱德思考试要求你熟练掌握幂函数、指数函数、对数函数和三角函数的导数。幂函数法则最常用。

• d/dx (xⁿ) = n xⁿ⁻¹
• d/dx (eˣ) = eˣ
• d/dx (ln x) = 1/x
• d/dx (sin x) = cos x
• d/dx (cos x) = -sin x

For fractional and negative powers, apply the same rule: d/dx (√x) = d/dx (x^½) = ½ x^-½ = 1/(2√x).

对于分数指数和负指数，使用相同法则：d/dx (√x) = d/dx (x^½) = ½ x^-½ = 1/(2√x)。

Always simplify coefficients before differentiating so the derivative is easier to work with.

求导前先化简系数，这样导数更容易处理。

3. Differentiation from first principles | 从第一原理求导

Some Edexcel questions ask you to prove the derivative of a simple function from first principles. You must show the limit of a difference quotient.

一些爱德思题目会要求你用第一原理证明简单函数的导数。你必须展示差商的极限过程。

For f(x) = x² from first principles, start with f(x + h) = (x + h)² = x² + 2xh + h².

对于 f(x) = x² 的第一原理，先写出 f(x + h) = (x + h)² = x² + 2xh + h²。

[(x + h)² – x²] / h = (2xh + h²) / h = 2x + h

As h → 0, the expression 2x + h tends to 2x, so f'(x) = 2x.

当 h → 0 时，表达式 2x + h 趋于 2x，因此 f'(x) = 2x。

Always expand the brackets, cancel terms, divide by h, and then let h tend to 0.

始终要展开括号、消去项、除以 h，然后让 h 趋于 0。

4. Chain rule, product rule and quotient rule | 链式法则、乘积法则与商法则

For composite functions, use the chain rule. For products and quotients, apply the product and quotient rules. These appear frequently in Papers 1 and 2.

对于复合函数，使用链式法则。对于乘积和商，使用乘积法则和商法则。这些在试卷一和二中常出现。

• Chain rule: dy/dx = dy/du × du/dx
• Product rule: d/dx (uv) = u dv/dx + v du/dx
• Quotient rule: d/dx (u/v) = (v du/dx – u dv/dx) / v²

For example, if y = (2x + 1)⁵, let u = 2x + 1 so y = u⁵. Then dy/du = 5u⁴ and du/dx = 2, giving dy/dx = 10(2x + 1)⁴.

例如，如果 y = (2x + 1)⁵，令 u = 2x + 1，则 y = u⁵。那么 dy/du = 5u⁴，du/dx = 2，因此 dy/dx = 10(2x + 1)⁴。

For y = x² sin x, use the product rule: dy

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