📚 Mastering Differentiation for Edexcel A-Level Maths | 攻克爱德思A-Level数学微分
Differentiation is the branch of calculus that studies how functions change. In the Edexcel A-Level Mathematics specification, it appears across Pure Mathematics, mechanics and applied contexts. Mastery of differentiation rules, stationary points and optimisation is essential for high marks.
微分是研究函数如何变化的微积分分支。在爱德思A-Level数学大纲中，它贯穿纯数学、力学和应用题。掌握微分法则、驻点和优化对于取得高分至关重要。
1. The Idea of a Derivative | 导数的思想
The derivative of a function f(x) at a point x=a gives the gradient of the tangent to the curve y=f(x) at that point. It is defined as the limit of the average rate of change over smaller and smaller intervals.
函数 f(x) 在 x=a 处的导数给出曲线 y=f(x) 在该点切线的斜率。它定义为当区间越来越小时平均变化率的极限。
For a straight line, gradient = Δy / Δx. For a curve, we take the limit as Δx → 0. This limiting value is written f'(a) or dy/dx at x=a.
对于直线，斜率 = Δy / Δx。对于曲线，我们取 Δx → 0 时的极限。这个极限值记作 f'(a) 或 x=a 时的 dy/dx。
2. Differentiation from First Principles | 从第一性原理求导
The formal definition of the derivative is:
导数的正式定义是：
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
This process is called differentiation from first principles. It underpins every standard derivative result used in the exam.
这个过程称为从第一性原理求导。它支撑着考试中使用的每一个标准导数结果。
Example: For f(x)=x², f(x+h)−f(x)=(x+h)²−x²=2xh+h². Dividing by h gives 2x+h. As h → 0, the limit is 2x, so f'(x)=2x.
示例：对于 f(x)=x²，f(x+h)−f(x)=(x+h)²−x²=2xh+h²。除以 h 得到 2x+h。当 h → 0 时，极限为 2x，所以 f'(x)=2x。
3. Standard Derivatives and the Power Rule | 标准导数与幂法则
The power rule states that if f(x)=xⁿ, then f'(x)=nxⁿ⁻¹. It works for any real constant n, including negative and fractional powers.
幂法则指出，如果 f(x)=xⁿ，那么 f'(x)=nxⁿ⁻¹。它适用于任何实数常数 n，包括负指数和分数指数。
Key standard derivatives include:
关键标准导数包括：
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导