📚 Question 11: Differentiation from First Principles | 第11题：从第一原理求导
In AQA A-Level Mathematics, Question 11 of the Pure Mathematics paper frequently assesses the technique of differentiating from first principles. This is one of the most accessible yet most frequently lost marks in the exam, because students often memorise the rules of differentiation without understanding how those rules are derived.
在 AQA A-Level 数学考试中，纯数试卷的第 11 题经常考查从第一原理求导的技巧。这是考试中最容易得分却又最常失分的题目之一，因为学生往往记住了求导法则，却不理解这些法则如何推导而来。
1. The Gradient of a Curve | 曲线上的斜率
f(x), we take a nearby point Q, draw the chord PQ, and then let Q slide closer and closer to P.
对于直线，斜率是恒定的，计算方法为 y 的变化量除以 x 的变化量。然而对曲线而言，斜率逐点变化。要找出曲线 y = f(x) 上某一点 P 的斜率，我们取邻近一点 Q，画出弦 PQ，然后令 Q 不断靠近 P。
Suppose P has x-coordinate x and Q has x-coordinate x + δx. The vertical distance between P and Q is:
假设 P 的横坐标为 x，Q 的横坐标为 x + δx。P 与 Q 之间的垂直距离为：
δy = f(x + δx) − f(x)
So the gradient of the chord PQ is:
因此弦 PQ 的斜率为：
δy / δx = [f(x + δx) − f(x)] / δx
As δx approaches 0, the chord becomes the tangent at P, and this gradient becomes the derivative of the function
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