Differentiation and Integration: Core Techniques and Applications | 微分与积分：核心技巧与应用

📚 Differentiation and Integration: Core Techniques and Applications | 微分与积分：核心技巧与应用

Differentiation and integration are the two central operations of calculus. A-level mathematics papers use them in a wide range of contexts, from finding gradients and turning points to calculating areas and solving optimisation problems. This article reviews the essential rules, common applications and exam techniques covered in many pure mathematics courses.

微分和积分是微积分的两个核心运算。A-level 数学试卷在各类情境中都会涉及它们，从求斜率和转折点到计算面积以及解决最优化问题。本文回顾纯数学课程中常见的基本法则、常见应用和考试技巧。

1. The Derivative and Its Meaning | 导数及其意义

The derivative measures the instantaneous rate of change of a function y = f(x) with respect to x. It is written as f'(x), dy/dx or d/dx[f(x)]. Geometrically, f'(a) is the gradient of the tangent to the curve y = f(x) at x = a.

导数衡量函数 y = f(x) 相对于 x 的瞬时变化率。它写作 f'(x)、dy/dx 或 d/dx[f(x)]。从几何上看，f'(a) 是曲线 y = f(x) 在 x = a 处切线的斜率。

The derivative is defined by the limit f'(x) = lim (h→0) [(f(x+h) – f(x))/h], provided this limit exists. If the limit does not exist at a point, the function is not differentiable there.

导数的定义是极限 f'(x) = lim (h→0) [(f(x+h) – f(x))/h]，前提是该极限存在。如果某点极限不存在，函数在该点不可导。

Common notations include dy/dx, f'(x), y’ and Leibniz notation d/dx. In A-level problems, you are often given y in terms of x and asked to find dy/dx.

常见记号包括 dy/dx、f'(x)、y’ 以及莱布尼茨记号 d/dx。在 A-level 题目中，通常会给出 y 关于 x 的表达式，要求求出 dy/dx。

2. Basic Differentiation Rules | 基本微分法则

For any constant c and any real number n, the power rule states d/dx[xⁿ] = n xⁿ⁻¹. This works for positive integers, negative powers and fractional powers.

对于任意常数 c 和实数 n，幂法则指出 d/dx[xⁿ] = n xⁿ⁻¹。这适用于正整数、负指数和分数指数。

The derivative of a constant is zero: d/dx[c] = 0. The constant multiple rule gives d/dx[c f(x)] = c f'(x). The sum and difference rules allow term-by-term differentiation.

常数的导数为零：d/dx[c] = 0。常数倍法则给出 d/dx[c f(x)] = c f'(x)。和差法则允许逐项求导。

Common derivatives to memorise: d/dx[sin x] = cos x, d/dx[cos x] = -sin x, d/dx[eˣ] = eˣ, d/dx[ln x] = 1/x for x > 0.

需要熟记的常见导数：d/dx[sin x] = cos x，d/dx[cos x] = -sin x，d/dx[eˣ] = eˣ，d/dx[ln x] = 1/x（x > 0）。

Before differentiating, rewrite expressions using index laws. For example, 1/x² becomes x⁻², and √

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