Edexcel A-Level Maths: Differentiation from First Principles | 爱德思 A-Level 数学：从第一性原理求导

📚 Edexcel A-Level Maths: Differentiation from First Principles | 爱德思 A-Level 数学：从第一性原理求导

In Edexcel A-Level Mathematics, differentiation is one of the most heavily examined topics in Pure Mathematics. Whether you are sitting AS Paper 1 or the full A-Level Paper 1, you need to be confident moving from a chord gradient to a derivative, applying the power rule, and interpreting gradients in tangents, normals and stationary points. Many exam questions also ask for a proof from first principles, so the limit definition must be known precisely.

在爱德思 A-Level 数学中，微分是纯数部分考查最频繁的主题之一。无论你参加 AS 试卷 1 还是完整 A-Level 试卷 1，都需要熟练地从割线斜率过渡到导数，应用幂法则，并在切线、法线和驻点问题中解释斜率。许多考试题还会要求从第一性原理进行证明，因此必须准确掌握极限定义。

1. The Gradient of a Curve as a Limit | 曲线斜率与极限

In coordinate geometry, a straight line has a constant gradient m = (y₂ − y₁)/(x₂ − x₁). For a curve such as y = x² or y = sin x, the steepness changes continuously, so a single gradient value cannot describe the whole curve. Instead, we describe the gradient at a point by using the slope of the tangent to the curve at that point.

在坐标几何中，直线具有恒定斜率 m=(y₂−y₁)/(x₂−x₁)。对于 y=x² 或 y=sin x 这样的曲线，陡峭程度不断变化，因此无法用单一斜率值描述整条曲线。相反，我们用曲线在该点处切线的斜率来描述该点的斜率。

To define the tangent rigorously, we start with a chord joining two close points. Let the first point be (x, f(x)) and the second point be (x+h, f(x+h)), where h is a small change in x. The chord gradient is the vertical change divided by the horizontal change.

为了严格定义切线，我们先从连接两个相邻点的割线开始。设第一个点为 (x, f(x))，第二个点为 (x+h, f(x+h))，其中 h 是 x 的微小变化。割线斜率等于竖直变化量除以水平变化量。

Chord gradient = [f(x+h) − f(x)] / h

割线斜率 = [f(x+h) − f(x)] / h

As h approaches 0, the second point slides towards the first, and the chord becomes the tangent. The limiting value of the chord gradient is called the derivative, written f'(x) or dy/dx.

当 h 趋近 0 时，第二个点滑向第一个点，割线变成切线。割线斜率的极限值称为导数，记作 f'(x) 或 dy/dx。

2. Differentiation from First Principles | 第一性原理求导

The first-principles formula is the foundation of differentiation. It is given by the following limit:

第一性原理公式是微分的基础。它由以下极限给出：

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

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

Edexcel frequently asks candidates to use this definition to prove standard derivatives, so you must show the expansion, subtraction and cancellation steps in full. Do not simply write the answer without showing the limit process.

爱德思考试经常要求学生使用这个定义证明标准导数，因此必须完整展示展开、相减和约分的步骤。不要只写出最终答案而不展示极限过程。

A common mistake is to

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