📚 Differentiation | 微分法
In Edexcel A-Level Mathematics, differentiation is a central topic in Pure Mathematics, appearing across Papers 1, 2 and 3. It is used to measure rates of change, to find gradients of curves, and to solve optimisation problems. A confident command of differentiation rules and applications is essential for high marks.
在 Edexcel A-Level 数学中，微分是纯数学的核心主题，出现在试卷 1、2 和 3 中。它用于测量变化率、求曲线梯度以及解决最优化问题。熟练掌握微分法则及其应用是取得高分的关键。
1. What Differentiation Means | 微分的含义
angent to the curve at that point. For a function y = f(x), the derivative is denoted by f'(x) or dy/dx.
微分是求函数瞬时变化率的过程。从几何上看，某一点的导数给出了曲线在该点处切线的斜率。对于函数 y = f(x)，导数记为 f'(x) 或 dy/dx。
In real-world contexts, differentiation helps model velocity, acceleration, marginal cost, and population growth rate. Understanding this concept is the foundation for many later topics in the A-Level syllabus.
在实际情境中，微分有助于建立速度、加速度、边际成本和人口增长率等模型。理解这一概念是 A-Level 课程中许多后续主题的基础。
2. First Principles | 从第一原理出发
The derivative is formally defined by a limit. The gradient of the chord from x to x + h approaches the gradient of the tangent as h tends to zero:
导数由极限正式定义。当 h 趋于零时，从 x 到 x + h 的弦的斜率趋近于切线的斜率：
f'(x) = limₕ→0 [ f(x + h) − f(x) ] / h
This expression is called differentiation from first principles. It is used in exam questions to prove basic derivative results, especially for simple powers of x.
这个表达式称为从第一原理出发求导。考试中常用它来证明基本的导数结果，尤其是 x 的简单幂函数。
For example, if f(x) = x², expanding f(x + h) = (x + h)² gives f'(x) = 2x after simplification and taking the limit.
例如，若 f(x) = x²，展开 f(x + h) = (x + h)²，化简并取极限后可得 f'(x) = 2x。
3. Basic Differentiation Rules | 基本微分法则
Before attempting complex functions, you must be fluent in the three basic rules. These allow you to differentiate polynomials and simple sums quickly.
在尝试复杂函数之前，你必须熟练掌握三条基本法则。这些法则能让你快速对多项式和简单和式求导。
- Constant rule: d/dx[c] = 0 — 常数法则：d/dx[c] = 0
- Power rule: d/dx[xⁿ] = n xⁿ⁻¹ — 幂法则：d/dx[xⁿ] = n xⁿ⁻¹
- Sum rule: d/dx[f(x) ± g(x)] = f'(x) ± g'(x) — 和法则：d/dx[f(x) ± g(x)] = f'(x) ± g'(x)
e, if y = 4x³ − 2x + 7, then dy/dx = 12x² − 2. The constant term 7 differentiates to zero.
例如，若 y = 4x³ − 2x + 7，则 dy/dx = 12x² − 2。常数项 7 的导数为零。
4. The Chain Rule | 链式法则
nother, such as y = (3x + 2)⁵. Write u = 3x + 2 and y = u⁵, then multiply the derivatives:
当一个函数嵌套在另一个函数内部时，例如 y = (3x + 2)⁵，需要使用链式法则。设 u = 3x + 2 且 y = u⁵，然后将各导数相乘：
dy/dx = dy/du × du/dx
⁴ and du/dx = 3, so dy/dx = 15(3x + 2)⁴. This method is essential for composite functions.
以上例来说，dy/du = 5u⁴，du/dx = 3，因此 dy/dx = 15(3x + 2)⁴。此方法对复合函数至关重要。
Always identify the inner function clearly. Common inner functions include linear terms, powers, and trigonometric arguments such as sin(2x) or cos(3x).
始终要清楚地识别内层函数。常见的内层函数包括线性项、幂以及三角函数的角度，例如 sin(2x) 或 cos(3x)。
5. Product Rule | 乘积法则
When a function is the product of two simpler functions, use the product rule. If u and v are both functions of x, then:
当一个函数是两个较简单函数的乘积时，使用乘积法则。若 u 和 v 都是 x 的函数，则：
d/dx[u v] = u' v + u v'
For example, if y = x² sin x, let u = x² and v = sin x. Then u' = 2x and v' = cos x, giving dy/dx = 2x sin x + x² cos x.
例如，若 y = x² sin x，设 u = x² 且 v = sin x。则 u' = 2x，v' = cos x，于是 dy/dx = 2x sin x + x² cos x。
Keep the functions in the original order in your final answer unless simplification is required. Many exam marks are lost through sign errors or missing derivatives.
除非需要化简，否则最终答案中保持函数原来的顺序。许多考试失分都是由于符号错误或遗漏导数造成的。
6. Quotient Rule | 商法则
For a function written as a fraction, use the quotient rule. If u and v are functions of x, and v ≠ 0, then:
对于以分式形式表示的函数，使用商法则。若 u 和 v 是 x 的函数，且 v ≠ 0，则：
d/dx[u / v] = (u' v − u v') / v²
For example, if y = x / (x + 1), let u = x and v = x + 1. Then u' = 1 and v' = 1, so dy/dx = [(1)(x + 1) − (x)(1)] / (x + 1)² = 1 / (x + 1)².
例如，若 y = x / (x + 1)，设 u = x，v = x + 1。则 u' = 1，v' = 1，所以 dy/dx = [(1)(x + 1) − (x)(1)] / (x + 1)² = 1 / (x + 1)²。
Remember that the numerator has u' v first and u v' second, then the whole expression is divided by the square of the denominator.
请记住，分子中先写 u' v，再写 u v'，然后将整个表达式除以分母的平方。
7. Differentiating Standard Functions | 标准函数的微分
Edexcel A-Level questions expect rapid recall of derivatives for exponential, logarithmic, and trigonometric functions:
Edexcel A-Level 考试要求快速回忆指数函数、对数函数和三角函数的导数：
- d/dx[eˣ] = eˣ
- d/dx[ln x] = 1 / x, for x > 0 — x > 0 时成立
- d/dx[sin x] = cos x
- d/dx[cos x] = −sin x
- d/dx[tan x] = sec² x
esults combine with the chain rule when the argument is not simply x. For example, d/dx[e²ˣ] = 2e²ˣ and d/dx[ln(5x)] = 1/x.
当自变量不只是 x 时，这些结果需要与链式法则结合使用。例如，d/dx[e²ˣ] = 2e²ˣ，d/dx[ln(5x)] = 1/x。
For sec x, cosec x and cot x, their derivatives are less commonly required but can be derived using the quotient rule if needed.
对于 sec x、cosec x 和 cot x，它们的导数较少直接考查，但需要时可用商法则推导。
8. Tangents and Normals | 切线与法线
get the gradient m = dy/dx at that point. Then substitute into the straight line formula:
要求某点处切线的方程，首先通过微分得到该点处的斜率 m = dy/dx。然后代入直线方程公式：
y − y₁ = m(x − x₁)
The normal is perpendicular to the tangent, so its gradient is −1/m. If the tangent has gradient 3, the normal has gradient −1/3.
法线与切线垂直，因此其斜率为 −1/m。如果切线斜率为 3，则法线斜率为 −1/3。
Always check whether the question asks for a tangent or a normal, and ensure the coordinates used satisfy the original curve equation.
一定要看清题目要求的是切线还是法线，并确保所用的坐标满足原曲线方程。
9. Stationary Points and Optimisation | 驻点与最优化
derivative d²y/dx². If d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum.
驻点出现在 dy/dx = 0 处。要判断其类型，需求二阶导数 d²y/dx²。若 d²y/dx² > 0，该点为局部极小值；若 d²y/dx² < 0，该点为局部极大值。
If the second derivative is zero, use a sign test on dy/dx around the point. This is especially important for cubic graphs and optimisation problems.
若二阶导数为零，则需在驻点附近对 dy/dx 进行符号检验。这对于三次函数图像和最优化问题尤其重要。
ntiate it, and solve dy/dx = 0 to find the value that maximises or minimises the quantity.
在应用题中，先建立体积或面积等表达式，对其求导，再解 dy/dx = 0，以找到使该量达到最大或最小的值。
10. Implicit and Parametric Differentiation | 隐函数与参数方程微分
espect to x, treating y as a function of x. Each time you differentiate y, multiply by dy/dx:
对于 x² + y² = 25 这样的隐式方程，需对每一项关于 x 求导，并将 y 视为 x 的函数。每次对 y 求导时都要乘以 dy/dx：
2x + 2y(dy/dx) = 0 ⇒ dy/dx = −x / y
= g(t), the gradient is found by dividing the derivatives with respect to the parameter t:
对于参数方程，若 x = f(t) 且 y = g(t)，则斜率通过将各自对参数 t 求导后相除得到：
dy/dx = (dy/dt) / (dx/dt)
This technique is common in questions involving circles, ellipses, and curves defined parametrically. Be careful with the signs of dx/dt and dy/dt.
此技巧常见于涉及圆、椭圆以及由参数定义的曲线的问题中。请特别注意 dx/dt 和 dy/dt 的符号。
11. Exam Technique and Common Errors | 考试技巧与常见错误
Strong differentiation skills are not just about knowing rules; they are about applying them under time pressure. The following strategies will help you secure full marks.
扎实的微分能力不仅在于掌握规则，还在于在时间压力下正确运用它们。以下策略将帮助你拿到满分。
- Always show your working clearly, even when using a calculator to check answers. — 始终清晰展示解题步骤，即使你用计算器核对答案。
- Do not confuse differentiating xⁿ with integrating xⁿ. — 不要把 xⁿ 的求导与积分混淆。
- Remember to use the chain rule when differentiating composite functions. — 对复合函数求导时，务必使用链式法则。
- Check whether a question requires an exact answer or a decimal approximation. — 检查题目要求精确答案还是近似小数。
- In optimisation, verify that your stationary point is a maximum or minimum as requested. — 在最优化问题中，验证驻点是否为题目要求的极大值或极小值。
By practising a variety of past paper questions, you will become quicker at recognising which rule applies to each given function.
通过练习各种历年真题，你将能够更快地判断每个给定函数适用哪一条法则。
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