Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分

📚 Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分

Differentiation is one of the most important tools in Edexcel A-Level Mathematics, linking Pure, Statistics and Mechanics. It allows you to calculate gradients, rates of change and turning points, and it is examined in many forms across all papers.

微分是 Edexcel A-Level 数学中最重要的工具之一，贯穿纯数学、统计与力学。它可用于计算梯度、变化率和驻点，并以多种形式出现在各试卷中。

1. First Principles and Definition | 导数定义与第一原理

The derivative f ‘(x) gives the exact gradient of the curve y = f(x) at a point. The formal definition uses the limit of a chord gradient as h tends to 0.

导数 f ‘(x) 给出曲线 y = f(x) 在某点处的精确梯度。其正式定义使用当 h 趋于 0 时弦梯度的极限。

f ‘(x) = limₕ→₀ [f(x+h) − f(x)] / h

To differentiate from first principles, substitute f(x+h) and f(x), simplify the difference quotient, then let h tend to 0.

要从第一原理求导，先代入 f(x+h) 和 f(x)，化简差商，再令 h 趋于 0。

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h². The difference quotient simplifies to 2x + h, so f ‘(x) = 2x as h → 0.

例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。差商化简为 2x + h，因此当 h → 0 时 f ‘(x) = 2x。

2. Power Rule and Basic Polynomials | 幂函数法则与多项式

For any real power n, the power rule states that d/dx (xⁿ) = n xⁿ⁻¹. This is the fastest way to differentiate polynomials and simple powers of x.

对任意实数幂 n，幂函数法则为 d/dx (xⁿ) = n xⁿ⁻¹。这是对多项式和 x 的简单幂求导的最快方法。

If f(x) = axⁿ where a is a constant, then f ‘(x) = n a xⁿ⁻¹. Constants disappear because their gradient is zero.

若 f(x) = axⁿ，其中 a 为常数，则 f ‘(x) = n a xⁿ⁻¹。常数项导数为零，因为其梯度为零。

For example, if y = 3x⁵ − 4x³ + 2x − 7, then dy/dx = 15x⁴ − 12x² + 2.

例如，若 y = 3x⁵ − 4x³ + 2x − 7，则 dy/dx = 15x⁴ − 12x² + 2。

Negative and fractional powers can be rewritten: d/dx (1/x) = d/dx (x⁻¹) = −x⁻² = −1/x².

负指数和分数指数可先改写：d/dx (1/x) = d/dx (x⁻¹) = −x⁻² = −1/x²。

3. Chain Rule | 链式法则

The chain rule is used when one function is inside another. If y = f(u) and u = g(x), then:

当函数套函数时使用链式法则。若 y = f(u) 且 u = g(x)，则：

dy/dx = dy/du × du/dx

In function notation, if y = f(g(x)), then dy/dx = f ‘(g(x)) × g ‘(x). This is often remembered as ‘differentiate the outside, keep the inside, then multiply by the derivative of the inside’.

用函数记号表示，若 y = f(g(x))，则 dy/dx = f ‘(g(x)) × g ‘(x)。这常被记为“先对外层求导，保留内层，再乘以内层的导数”。

For example,

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