Differentiation: Rules and Applications | 微分：法则与应用

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

Differentiation is a cornerstone of A-Level mathematics, enabling us to measure how a function changes at any given point. In the AQA specification, differentiation underpins topics from coordinate geometry to kinematics, and it is frequently examined as Question 4 in both pure and applied papers.

微分是A-Level数学的基石，它使我们能够度量函数在任意给定点的变化情况。在AQA考纲中，微分支撑着从坐标几何到运动学等多个主题，并且经常作为第4题出现在纯数学和应用数学试卷中。

1. Understanding Differentiation from First Principles | 从基本原理理解微分

The derivative is formally defined as the limit of the average rate of change as the interval approaches zero. For a function f(x), the derivative f'(x) is given by the following expression:

导数的正式定义是当区间趋近于零时平均变化率的极限。对于函数f(x)，导数f'(x)由以下表达式给出：

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

This definition is known as differentiation from first principles. In the AQA examination, you may be asked to apply this definition to simple polynomial functions, such as f(x) = x² or f(x) = x³.

这个定义被称为从基本原理出发的微分。在AQA考试中，可能会要求你将此定义应用于简单的多项式函数，如f(x) = x²或f(x) = x³。

For example, if f(x) = x², then f(x+h) = (x+h)² = x² + 2xh + h². Substituting into the definition:

例如，若f(x) = x²，则f(x+h) = (x+h)² = x² + 2xh + h²。将其代入定义：

f'(x) = lim(h→0) [(x² + 2xh + h²) − x²] / h = lim(h→0) (2x + h) = 2x

As h approaches zero, the term 2x + h approaches 2x, giving us the familiar result. This process forms the foundation of all calculus and is worth practising thoroughly.

当h趋近于零时，2x + h这一项趋近于2x，从而得到我们熟悉的结果。这个过程构成了所有微积分的基础，值得反复练习。

2. The Basic Rules of Differentiation |

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