📚 Essential Differentiation and Integration for Edexcel A-Level Pure Maths | Edexcel A-Level 纯数学微分与积分核心要点
Differentiation and integration are the twin engines of Edexcel A-Level Pure Mathematics. They dominate Paper 1 and Paper 2, and they also underpin mechanics and statistics applications. This revision guide summarises the highest-yield techniques, from first principles to parametric rates of change, with exam-style commentary.
微分和积分是 Edexcel A-Level 纯数学的两大核心引擎。它们不仅主导 Paper 1 和 Paper 2，还是力学和统计应用的基础。本复习指南总结了最高频的解题技巧，从第一性原理到参数变化率，并配有考试风格点评。
1. First Principles and Notation | 第一性原理与记号
The derivative of f(x) is defined by the limit f'(x) = limₕ→₀ [f(x+h) – f(x)]/h. In Edexcel exams, you may be asked to use this definition for simple functions such as x², x³ or 1/x.
f(x) 的导数由极限 f'(x) = limₕ→₀ [f(x+h) – f(x)]/h 定义。在 Edexcel 考试中，你可能需要用这个定义来求简单函数，如 x²、x³ 或 1/x 的导数。
You must be comfortable with notation: dy/dx, f'(x), and ẏ when time t is the variable. Misreading notation is a common source of lost marks in mechanics questions.
你必须熟悉多种记号：dy/dx、f'(x)，以及以时间 t 为变量时的 ẏ。在力学题目中，误读记号是常见的失分原因。
The core standard derivatives are:
核心标准导数如下：
| f(x) | f'(x) |
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
2. Product, Quotient and Chain Rules | 乘法、商法和链式法则
The product rule states d/dx(uv) = u dv/dx + v du/dx. The quotient rule states d/dx(u/v) = (v du/dx – u dv/dx)/v². The chain rule states dy/dx = dy/du × du/dx.
乘法法则为 d/dx(uv) = u dv/dx + v du/dx。商法法则为 d/dx(u/v) = (v du/dx – u dv/dx)/v²。链式法则为 dy/dx = dy/du × du/dx。
In practice, choose u and v strategically to make the resulting derivative simpler. For chain rule, look for an inner function and multiply by its derivative.
实际解题时，要策略性地选择 u 和 v，使后续导数更简单。对于链式法则，找出内层函数并乘以其导数。
For example, if y = (2x + 1)³, set u = 2x + 1, so y = u³. Then dy/du = 3u² and du/dx = 2, giving dy/dx = 6(2x + 1)².
例如，如果 y = (2x + 1)³，设 u = 2x + 1，则 y = u³。那么 dy/du = 3u²，du/dx = 2，因此 dy/dx = 6(2x + 1)²。
3. Standard Derivatives and Second Derivatives | 标准导数与二阶导数
Be fluent with derivatives of eˣ, ln x, sin x, cos x, tan x, and powers of x. The second derivative f”(x) tells you about concavity and helps classify stationary points.
要熟练掌握 eˣ、ln x、sin x、cos x、tan x 和 x 的幂的导数。二阶导数 f”(x) 表示凹凸性，并有助于判断驻点的性质。
For example, if y = 3x⁴ – 2eˣ + 5 sin x, then dy/dx = 12x³ – 2eˣ + 5 cos x, and d²y/dx² = 36x² – 2eˣ – 5 sin x.
例如，如果 y = 3x⁴ – 2eˣ + 5 sin x，那么 dy/dx = 12x³ – 2eˣ + 5 cos x，且 d²y/dx² = 36x² – 2eˣ – 5 sin x。
Extended questions often combine differentiation with trigonometric identities. Always simplify before differentiating when possible, such as turning tan x into sin x / cos x if needed.
扩展题常将微分与三角恒等式结合。尽可能先化简再求导，例如需要时将 tan x 写成 sin x / cos x。
4. Tangents, Normals, Increasing and Decreasing Functions | 切线、法线、递增递减函数
The tangent at a point (a, f(a)) has equation y – f(a) = f'(a)(x – a). The normal has gradient -1/f'(a). A function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.
在点 (a, f(a)) 处的切线方程为 y – f(a) = f'(a)(x – a)。法线的斜率为 -1/f'(a)。当 f'(x) > 0 时函数递增，当 f'(x) < 0 时递减。
Exam questions often require you to find a normal to a curve and then find where it crosses the axes. Always write the equation in the required form ax + by + c = 0.
考试题常要求你求出曲线的法线，然后求出它与坐标轴的交点。始终将方程写成题目要求的 ax + by + c = 0 形式。
For example, the normal to y = x² at x = 1 has gradient -1/2 and passes through (1, 1), so its equation is y – 1 = -1/2(x – 1), or x + 2y – 3 = 0.
例如，曲线 y = x² 在 x = 1 处的法线斜率为 -1/2，且经过 (1, 1)，因此其方程为 y – 1 = -1/2(x – 1)，即 x + 2y – 3 = 0。
5. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where f'(x) = 0. Use f”(x) or a sign table to classify them as maximum, minimum, or point of inflection.
驻点出现在 f'(x) = 0 处。用 f”(x) 或符号表将其分类为极大值、极小值或拐点。
When sketching, identify intercepts, asymptotes, stationary points and behaviour as x → ±∞. Marking schemes reward a clear shape and labelled key features.
画草图时，要确定截距、渐近线、驻点以及 x → ±∞ 时的变化趋势。评分方案会给清晰的形状和标注的关键特征加分。
A maximum has f”(x) < 0, a minimum has f”(x) > 0, and if f”(x) = 0 you should check either side of the point using a sign table.
极大值处 f”(x) < 0，极小值处 f”(x) > 0，如果 f”(x) = 0，则应使用符号表检查该点两侧的情况。
6. Integration as Reverse Differentiation | 积分作为微分的逆运算
Indefinite integration reverses differentiation: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C for n ≠ -1. The constant C is essential in Edexcel mark schemes.
不定积分是微分的逆运算：当 n ≠ -1 时，∫ xⁿ dx = xⁿ⁺¹/(n+1) + C。常数 C 在 Edexcel 评分标准中至关重要。
You should also know ∫ eˣ dx = eˣ + C, ∫ 1/x dx = ln|x| + C, ∫ cos x dx = sin x + C, and ∫ sin x dx = -cos x + C.
你还需要掌握 ∫ eˣ dx = eˣ + C，∫ 1/x dx = ln|x| + C，∫ cos x dx = sin x + C，以及 ∫ sin x dx = -cos x + C。
Initial conditions are used to find C. For instance, if dy/dx = 3x² and y = 4 when x = 1, then y = x³ + 3.
初始条件用于求出常数 C。例如，若 dy/dx = 3x² 且 x = 1 时 y = 4，则 y = x³ + 3。
7. Integration by Substitution | 换元积分法
For integrals of the form ∫ f(g(x))g'(x) dx, set u = g(x) so du = g'(x) dx, and rewrite the integral as ∫ f(u) du.
对于形如 ∫ f(g(x))g'(x) dx 的积分，令 u = g(x)，则 du = g'(x) dx，将积分改写为 ∫ f(u) du。
For definite integrals, change the limits to u-values or return to x before substituting the original limits. Show your substitution clearly to gain method marks.
对于定积分，要把
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