Differentiation Rules and Applications | 微分法则与应用

📚 Differentiation Rules and Applications | 微分法则与应用

Differentiation is one of the most important tools in A-Level Mathematics. It measures how a function changes as its input changes, giving the gradient of a curve at any point. In the Edexcel specification, you need to differentiate standard functions, apply the chain, product and quotient rules, and use derivatives to solve problems involving tangents, normals, stationary points, rates of change and optimisation.

微分是 A-Level 数学中最重要的工具之一。它衡量函数随自变量变化的快慢，给出曲线上任意一点的斜率。在 Edexcel 考试大纲中，你需要对标准函数求导，应用链式法则、乘法法则和商法则，并利用导数解决切线、法线、驻点、变化率和优化等问题。

1. From First Principles | 从第一性原理出发

The derivative is defined by a limit. For a function y = f(x), the derivative at any point is the limit of the gradient of a chord between two points as they become arbitrarily close. This definition is important in Edexcel P1 and P2, and it can be used to prove standard results.

导数由极限定义。对于函数 y = f(x)，任意一点的导数是两点间弦的斜率在两点无限接近时的极限。这个定义在 Edexcel P1 和 P2 中都很重要，它可以用来证明标准结果。

f ‘ (x) = lim (h→0) [f(x+h) − f(x)] / h

For example, if f(x) = x², expand f(x+h) = (x+h)² = x² + 2xh + h². Then [f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h. As h → 0, the limit is 2x.

例如，若 f(x) = x²，展开 f(x+h) = (x+h)² = x² + 2xh + h²。于是 [f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h。当 h → 0 时，极限为 2x。

2. Standard Derivatives and Notation | 标准导数与记号

You should learn the following standard derivatives by heart. There are three common notations: f'(x), dy/dx, and d/dx [f(x)]. They all mean the same thing. The table below summarises the key results for Edexcel.

你应该熟记以下标准导数。常见记号有三种：f'(x)、dy/dx 和 d/dx [f(x)]。它们含义相同。下表总结了 Edexcel 的关键结果。

Note that the power rule works for any real constant n, including negative and fractional powers. For example, d/dx (√x) = 1/(2√x), since √x is the same as x^½.

注意幂法则适用于任何实数 n，包括负指数和分数指数。例如，d/dx (√x) = 1/(2√x)，因为 √x 等同于 x^½。

3. The 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 dy/dx = dy/du × du/dx.

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