📚 Differentiation Rules and Applications | 微分法则与应用
Differentiation is one of the most important tools in A-Level Mathematics. It measures how a function changes as its input changes, giving the gradient of a curve at any point. In the Edexcel specification, you need to differentiate standard functions, apply the chain, product and quotient rules, and use derivatives to solve problems involving tangents, normals, stationary points, rates of change and optimisation.
微分是 A-Level 数学中最重要的工具之一。它衡量函数随自变量变化的快慢，给出曲线上任意一点的斜率。在 Edexcel 考试大纲中，你需要对标准函数求导，应用链式法则、乘法法则和商法则，并利用导数解决切线、法线、驻点、变化率和优化等问题。
1. What Differentiation Measures | 微分的含义与考试范围
gent to the curve at that point. If x increases by a small amount h, the average rate of change is [f(x+h) − f(x)] / h, and the derivative is the limit of this expression as h → 0.
函数在某一点的导数是该点处的瞬时变化率。从几何上看，它是曲线在该点处切线的斜率。若 x 增加一个很小的量 h，平均变化率为 [f(x+h) − f(x)] / h，而导数就是当 h → 0 时该表达式的极限。
Edexcel expects you to differentiate standard functions confidently, use the chain, product and quotient rules, and then apply derivatives to tangents, normals, stationary points, rates of change and optimisation. This means differentiation is not only a set of rules but also a problem-solving toolkit.
Edexcel 要求你能够熟练地对标准函数求导，使用链式法则、乘法法则和商法则，并把导数应用到切线、法线、驻点、变化率和优化问题中。这意味着微分不仅是一组法则，更是一套解题工具。
Three common notations are used: f'(x), dy/dx and d/dx [f(x)]. They all represent the derivative. In written answers, choose one notation and use it consistently, especially when a question involves both a function and its derivative.
常见记号有三种：f'(x)、dy/dx 和 d/dx [f(x)]。它们都表示导数。在书面作答时，选择一种记号并保持一致，尤其是在题目同时涉及函数和导数时。
2. From First Principles | 从第一性原理出发
The formal definition of the derivative from first principles is based on the limit of the gradient of a chord between two points as they become arbitrarily close. This definition is important in Edexcel P1 and P2, and it can be used to prove standard results such as the derivative of x².
导数的第一性原理定义基于两点间弦的斜率在两点无限接近时的极限。这个定义在 Edexcel P1 和 P2 中都很重要，它可以用来证明标准结果，例如 x² 的导数。
f'(x) = lim (h→0) [f(x+h) − f(x)] / h
. Therefore [f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h. As h → 0, the limit is 2x, so f'(x) = 2x.
例如，若 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。于是 [f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h。当 h → 0 时，极限为 2x，所以 f'(x) = 2x。
ully, cancel h where possible, and then let h → 0. Do not divide by h before simplifying the numerator.
第一性原理题通常要求你证明某个结果，或对简单的多项式、有理函数求导。关键是仔细展开，在可能的地方约去 h，然后令 h → 0。在化简分子之前不要直接除以 h。
3. Standard Derivatives and Notation | 标准导数与记号
You should learn the following standard derivatives by heart. They appear throughout A-Level Mathematics and are the building blocks for the chain, product and quotient rules. The table below summarises the key results for Edexcel.
你应该熟记以下标准导数。它们贯穿 A-Level 数学，是链式法则、乘法法则和商法则的基础。下表总结了 Edexcel 的关键结果。
| f(x) | f'(x) |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
| aˣ | aˣ ln a |
The power rule works for any real constant n, including negative and fractional powers, provided the function is defined at the point. For example, d/dx (√x) = 1/(2√x), because √x is the same as x1/2. Using the power rule gives (1/2)x−1/2, which simplifies to 1/(2√x).
幂法则适用于任何实数常数 n，包括负指数和分数指数，前提是函数在该点有定义。例如，d/dx (√x) = 1/(2√x)，因为 √x 等同于 x1/2。使用幂法则得到 (1/2)x−1/2，化简后为 1/(2√x)。
lways check the domain when differentiating fractional or negative powers. For instance, x1/2 for x ≥ 0, and 1/x is not defined at x = 0. The derivative inherits similar restrictions.
对分数指数或负指数求导时，一定要检查定义域。例如，x1/2 只在 x ≥ 0 时有定义，而 1/x 在 x = 0 处无定义。导数也会继承类似的限制。
4. The Chain Rule | 链式法则
s, differentiate the outer function, then multiply by the derivative of the inner function.
链式法则用于一个函数套在另一个函数内部的情况，也就是复合函数。若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。换句话说，先对外层函数求导，再乘以内层函数的导数。
dy/dx = dy/du × du/dx
x = 5u⁴ × 6x = 30x(3x² + 1)⁴. This shows why the inner derivative must never be forgotten.
例如，对 y = (3x² + 1)⁵ 求导。令 u = 3x² + 1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x。因此 dy/dx = 5u⁴ × 6x = 30x(3x² + 1)⁴。这说明为什么绝不能忘记内层导数。
nential functions, the same idea applies. If y = sin(2x), then dy/dx = 2 cos(2x). If y = ex², then dy/dx = 2x ex². If y = ln(5x + 1), then dy/dx = 5/(5x + 1). Always identify the inner function first.
对于三角函数和指数函数，同样适用。若 y = sin(2x)，则 dy/dx = 2 cos(2x)。若 y = ex²，则 dy/dx = 2x ex²。若 y = ln(5x + 1)，则 dy/dx = 5/(5x + 1)。一定要先识别内层函数。
5. The Product Rule | 乘法法则
e both functions of x, then dy/dx = u dv/dx + v du/dx. An equivalent form is d/dx (uv) = u'v + uv'. The rule is often remembered as "first times derivative of second plus second times derivative of first".
乘法法则用于两个函数相乘的情况。若 y = uv，其中 u 和 v 都是 x 的函数，则 dy/dx = u dv/dx + v du/dx。等价形式为 d/dx (uv) = u'v + uv'。通常记作 "第一项乘第二项的导数，加上第二项乘第一项的导数"。
d/dx (uv) = u dv/dx + v du/dx
cos x + 2x sin x. You can factorise it as x(x cos x + 2 sin x) if the question asks for a simplified form.
例如，对 y = x² sin x 求导。令 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x。因此 dy/dx = x² cos x + 2x sin x。如果
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