Differentiation from First Principles and Applications | 从第一性原理到应用：微分核心技巧

📚 Differentiation from First Principles and Applications | 从第一性原理到应用：微分核心技巧

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, candidates must be able to differentiate from first principles, use standard derivative rules, find tangents and normals, classify stationary points, and apply differentiation to real-world modelling problems.

微分是 A-Level 数学中最重要的工具之一。它衡量函数随输入变化而变化的速度，并给出曲线上任意一点的梯度。在 Edexcel 考试大纲中，考生必须掌握从第一性原理求导、使用标准求导法则、求切线与法线、判断驻点类型，以及将微分应用于实际建模问题。

1. The Gradient of a Curve and the Difference Quotient | 曲线梯度与差商

A straight line has a constant gradient, but a curve does not. To estimate the gradient of y = f(x) at a point x = a, we take a nearby point x = a + h and calculate the average rate of change between the two points.

直线具有恒定梯度，但曲线不是。为了估计 y = f(x) 在点 x = a 处的梯度，我们取附近一点 x = a + h，并计算两点之间的平均变化率。

Average gradient = [ f(a + h) − f(a) ] ÷ h

This expression is called the difference quotient, or the gradient of the chord joining the two points on the curve. As h becomes smaller, the chord gets closer to the tangent at x = a.

这个表达式称为差商，也就是曲线上两点之间弦的梯度。随着 h 越来越小，弦越来越接近 x = a 处的切线。

2. Differentiation from First Principles | 从第一性原理出发求导

The derivative of f(x) is defined as the limit of the difference quotient as h tends to 0. In Edexcel papers, you may be asked to prove the derivative of a simple function such as x², x³ or 1/x using this definition.

f(x) 的导数定义为差商在 h 趋向于 0 时的极限。在 Edexcel 试卷中，可能会要求用该定义证明 x²、x³ 或 1/x 等简单函数的导数。

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

For example, if f(x) = x², then f(x + h) = (x + h)² = x² + 2xh + h². Subtract f(x) = x², divide by h, and let h → 0. The result is 2x.

例如，若 f(x) = x²，则 f(x + h) = (x + h)² = x² + 2xh + h²。减去 f(x) = x²，再除以 h，并令 h → 0，结果为 2x。

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