📚 A-Level Edexcel Mathematics: Differentiation and Integration Essentials | A-Level Edexcel 数学：微分与积分核心精讲
Differentiation and integration are two of the most heavily weighted topics in the Edexcel A-Level Mathematics Pure papers. They appear not only as direct skill questions but also in modelling, optimisation, kinematics and area problems. This guide brings together the key facts, worked-style summaries and exam strategies to help you revise efficiently.
微分与积分是 Edexcel A-Level 数学纯数试卷中权重最高的两大主题。它们不仅以直接计算题出现，还会融入建模、优化、运动学和面积问题。本指南汇集关键知识点、解题思路与考试策略，帮助你高效复习。
1. The Derivative as a Gradient Function | 导数作为斜率函数
The derivative of y = f(x), written f'(x) or dy/dx, gives the gradient of the tangent to the curve at each point x. It is defined by the limit of the average rate of change as h tends to 0.
函数 y = f(x) 的导数，记作 f'(x) 或 dy/dx，表示曲线在每一点 x 处切线的斜率。它由平均变化率在 h 趋于 0 时的极限定义。
f'(x) = lim(h→0) [f(x+h) – f(x)]/h
Geometrically, the derivative tells you how steep the curve is at a point. If f'(x) > 0 the function is increasing at that point; if f'(x) < 0 the function is decreasing.
从几何上看，导数告诉你曲线在某一点的陡峭程度。如果 f'(x) > 0，函数在该点递增；如果 f'(x) < 0，函数在该点递减。
2. Differentiation Rules for Polynomials | 多项式的求导法则
For any real power n, the power rule states that the derivative of xⁿ is nxⁿ⁻¹. You can differentiate term by term, and constants can be carried through.
对于任意实数指数 n，幂法则指出 xⁿ 的导数是 nxⁿ⁻¹。你可以逐项求导，常数可以保留在求导过程中。
d/dx (xⁿ) = nxⁿ⁻¹
If f(x) = 4x³ – 2x + 7, then f'(x) = 12x² – 2. The derivative of a constant is 0.
如果 f(x) = 4x³ – 2x + 7，则 f'(x) = 12x² – 2。常数的导数为 0。
- d/dx [af(x) ± bg(x)] = af'(x) ± bg'(x)
- 中文：线性组合的
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