Mastering Differentiation for Edexcel A-Level Pure Maths | 掌握 Edexcel A-Level 纯数学中的微分

📚 Mastering Differentiation for Edexcel A-Level Pure Maths | 掌握 Edexcel A-Level 纯数学中的微分

Differentiation allows us to move from a function that describes position, area or total cost to a function that describes rate of change. In the Edexcel A-Level Pure Mathematics syllabus, differentiation appears in Pure Paper 1 and Paper 2, and it underpins many problems in mechanics and optimisation. This article revises the core techniques, notation and applications, with paired English and Chinese explanations.

微分使我们能够从描述位置、面积或总成本的函数转向描述变化率的函数。在 Edexcel A-Level 纯数学大纲中，微分出现在 Pure Paper 1 和 Paper 2 中，并为力学和最优化中的许多问题奠定基础。本文复习核心技术、记号和应用，并给出中英文对照解释。

1. What Differentiation Measures | 微分衡量什么

A straight line has a constant gradient. A curve does not have a single gradient, but we can still ask how steep the curve is at a particular point. The derivative dy/dx is the gradient of the tangent to the curve at that point.

直线具有恒定的斜率。曲线没有单一的斜率，但我们仍然可以考察曲线在某一点的陡峭程度。导数 dy/dx 就是曲线在该点的切线斜率。

If a curve represents displacement, the derivative represents velocity. If a curve represents cost, the derivative represents marginal cost. In pure mathematics, differentiation is the main tool for analysing local behaviour of functions.

如果曲线表示位移，则导数表示速度。如果曲线表示成本，则导数表示边际成本。在纯数学中，微分是分析函数局部行为的主要工具。

2. The First Principles Definition | 第一原理定义

The derivative is defined by a limit. For y = f(x), the gradient of the chord joining x and x + h is (f(x + h) − f(x)) / h.

导数由一个极限定义。对于 y = f(x)，连接 x 与 x + h 的弦的斜率为 (f(x + h) − f(x)) / h。

f'(x) = lim(h → 0) (f(x + h) − f(x)) / h

As h approaches 0, the chord becomes the tangent, and this limit gives the gradient of the curve at x. For example, if f(x) = x², expanding (x + h)² gives x² + 2xh + h², so the limit is 2x.

当 h 趋近于 0 时，弦变为切线，这个极限给出曲线在 x 处的斜率。例如，若 f(x) = x²，展开 (x + h)² 得到 x² + 2xh + h²，因此极限为 2x。

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