📚 Mastering Differentiation and Integration for Edexcel A-Level Pure Maths | 精通微分与积分：Edexcel A-Level 纯数学
Calculus is one of the most heavily weighted areas in Edexcel A-Level Pure Mathematics. It connects algebra, trigonometry and graphs, and appears across Paper 1 and Paper 2. Differentiation tells you how fast something changes; integration tells you how much has accumulated. This article reviews the key rules, the standard applications and the exam strategies that turn a shaky topic into reliable marks.
微积分是 Edexcel A-Level 纯数学中分值最高的模块之一。它连接代数、三角函数和图像，出现在 Paper 1 和 Paper 2 中。微分告诉你变化的快慢，积分告诉你累积的总量。本文将复习核心法则、标准应用，以及在考试中把这一薄弱环节变成稳定得分点的策略。
1. Where Calculus Sits in the Edexcel Specification | 微积分在 Edexcel 考纲中的位置
The specification treats differentiation and integration as two linked topic areas. The early material covers gradients, first principles, the power rule and simple polynomial curves. Later material extends to composite functions, trigonometric and exponential functions, rates of change, and the reverse process of integration.
考纲把微分与积分当作两个相互关联的主题板块。前期内容涵盖斜率、第一原理、幂法则和简单的多项式曲线；后期则延伸到复合函数、三角函数与指数函数、变化率，以及作为逆运算的积分。
Broadly speaking, Paper 1 tests the foundations: differentiating powers of x, finding tangents and normals, and locating stationary points. Paper 2 pushes further, with the chain, product and quotient rules, trigonometric and exponential functions, definite integrals, areas and the trapezium rule.
大致来说，Paper 1 考查基础：对 x 的幂函数求导、求切线与法线、求驻点；Paper 2 则更进一步，涉及链式、乘积与商法则、三角函数与指数函数、定积分、面积以及梯形法则。
Marks here are usually structured: a differentiation mark, a substitution mark, and an accuracy mark. That means method marks are available even when the final answer is wrong, so always write every step down in full.
这里的分值往往是分步给出的：求导分、代入分、准确分。这意味着即使最终答案错误，方法分仍然可得，所以一定要把每一步都完整写出来。
2. Gradient of a Curve and First Principles | 曲线的斜率与第一原理
The derivative of a function f(x) is defined as the limit of the average rate of change over an interval as the interval width tends to zero. This is known as differentiation from first principles.
函数 f(x) 的导数定义为区间宽度趋近于零时平均变化率的极限。这就是第一原理求导。
f′(x) = limₕ→₀ [ f(x + h) − f(x) ] / h
The numerator is the change in the value of the function and h is the change in x. The whole expression is called the difference quotient, and the derivative is its limit as h tends to zero.
分子是函数值的变化量，h 是 x 的变化量。整个表达式称为差商，导数就是当 h 趋于零时差商的极限。
h². Dividing by h gives 2x + h, and the limit as h tends to 0 is 2x. Therefore f′(x) = 2x.
对于简单二次函数 f(x) = x²，分子变为 (x + h)² − x² = 2xh + h²。除以 h 得到 2x + h，当 h 趋于 0 时极限为 2x。因此 f′(x) = 2x。
In Edexcel exams, first principles questions usually ask you to prove a standard result such as the derivative of x², x³ or √x. You must show the limit notation and simplify the difference quotient fully.
在 Edexcel 考试中，第一原理题通常要求证明标准结果，例如 x²、x³ 或 √x 的导数。你必须展示极限符号，并完整化简差商。
Two technical points earn marks. First, you may cancel h only because h is non-zero before the limit is taken, and you should say so. Second, when the function involves a square root, multiply the numerator and denominator by the conjugate √x + h + √x to remove the surd from the numerator.
有两个技术细节可以拿分。第一，只有在取极限之前 h 不为零时才能约去 h，并且要把这一点写出来。第二，当函数含根号时，用共轭式 √x + h + √x 同时乘分子与分母，把根号从分子中消去。
d(√x)/dx = 1 / (2√x)
should always state the limit clearly, for example: as h → 0, 2x + h → 2x, so f′(x) = 2x.
第一原理证明的最后一行一定要清楚写出极限，例如：当 h → 0 时，2x + h → 2x，所以 f′(x) = 2x。
3. The Power Rule and Differentiating Polynomials | 幂法则与多项式求导
For any real power n, the derivative of xⁿ is nxⁿ⁻¹. This single rule covers almost every polynomial and every expression that can be rewritten in index form.
对任意实数指数 n，xⁿ 的导数是 nxⁿ⁻¹。这一条法则几乎覆盖了所有多项式，以及一切可以改写为指数形式的表达式。
y = xⁿ ⇒ dy/dx = nxⁿ⁻¹
The most common exam trap is failing to rewrite roots and reciprocals as powers before differentiating. Marks are often awarded simply for the conversion step, so do it explicitly.
最常见的考试陷阱是：在求导之前没有把根式和倒数改写成幂的形式。这一转换步骤本身常常就有分数，所以要明确写出来。
| Function 函数 | Derivative 导数 | Note 说明 |
| y = x⁵ | dy/dx = 5x⁴ | power rule 幂法则 |
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