📚 Differentiation | 微分
Differentiation is one of the most important tools in Edexcel A-Level Mathematics. It measures how a function changes as its input changes, and it forms the basis for optimisation, curve sketching, kinematics and many applied problems. In this article, we review the core differentiation techniques, key formulas, and exam-style interpretation of derivatives.
微分是 Edexcel A-Level 数学中最重要的工具之一。它衡量函数随自变量变化而变化的速率，是优化、曲线作图、运动学以及许多应用问题的基础。本文回顾核心微分技巧、关键公式以及考试中导数的解释。
1. What is Differentiation? | 什么是微分？
For a function y = f(x), the derivative dy/dx gives the gradient of the tangent to the curve at any point x. It is defined as the limit of the average rate of change as the interval h tends to zero.
对于函数 y = f(x)，导数 dy/dx 给出曲线在任意点 x 处切线的斜率。它定义为当区间 h 趋于零时平均变化率的极限。
dy/dx = lim (h→0) [f(x+h) − f(x)] / h
The notation dy/dx is read as ‘the derivative of y with respect to x’. It can also be written as f'(x) or y’.
符号 dy/dx 读作 ‘y 对 x 的导数’。它也可以写成 f'(x) 或 y’。
2. First Principles | 从第一性原理求导
The first-principles definition is often examined to prove the derivative of x² or a simple polynomial. For f(x) = x², expanding f(x+h) − f(x) = (x+h)² − x² = 2xh + h² gives the result below.
第一性原理定义在考试中常用来证明 x² 或简单多项式的导数。对于 f(x) = x²，展开 f(x+h) − f(x) = (x+h)² − x² = 2xh + h²，可以得到以下结果。
dy/dx = lim (h→0) [2x + h] = 2x
This shows that the gradient of y = x² is exactly 2x at every point. The method requires simplifying the difference quotient before letting h approach zero.
这表明 y = x² 的斜率在每一点恰好为 2x。该方法需要先化简差商，再让 h 趋于零。
3. Standard Derivatives and Rules | 标准导数与法则
Memorising standard derivatives saves time. For any constant k, d/dx(k) = 0. For power functions, d/dx(xⁿ) = nxⁿ⁻¹. This rule extends to negative and fractional powers such as x⁻² and √x.
熟记标准导数可以节省时间。对任意常数 k，d/dx(k) = 0。对于幂函数，d/dx(xⁿ) = nxⁿ⁻¹。该法则可推广到负指数和分数指数，如 x⁻² 和 √x。
| Function | Derivative |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| sec x | sec x tan x |
| cosec x | −cosec x cot x |
| cot x | −cosec² x |
The linearity rule also applies: d/dx[a f(x) + b g(x)] = a f'(x) + b g'(x), where a and b are constants.
线性法则同样适用：d/dx[a f(x) + b g(x)] = a f'(x) + b g'(x)，其中 a 和 b 是常数。
4. Chain Rule | 链式法则
If y = g(u) and u = f(x), then dy/dx = dy/du × du/dx. This is used for composite functions such as y = (3x² + 5)⁴. Let u = 3x² + 5, then dy/du = 4u³ and du/dx = 6x, so dy/dx = 24x(3x² + 5)³.
如果 y = g(u) 且 u = f(x)，则 dy/dx = dy/du × du/dx。它用于复合函数，例如 y = (3x² + 5)⁴。设 u = 3x² + 5，则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 24x(3x² + 5)³。
You can also apply the chain rule quickly by differentiating the outer function and multiplying by the derivative of the inner function. For y = e^(2x), the derivative is 2e^(2x).
你也可以快速应用链式法则：先对外层函数求导，再乘以内层函数的导数。对于 y = e^(2x)，其导数为 2e^(2x)。
5. Product Rule | 乘积法则
When y = u(x)v(x), the product rule states dy/dx = u’v + uv’. This is useful for functions like y = x² eˣ and y = x sin x. A common mistake is forgetting to differentiate both factors or misapplying the chain rule inside u or v.
当 y = u(x)v(x) 时，乘积法则为 dy/dx = u’v + uv’。这对 y = x² eˣ 和 y = x sin x 等函数非常有用。常见错误：忘记对两个因式分别求导，或在 u 或 v 内部误用链式法则。
For y = x² eˣ, let u = x² and v = eˣ. Then u’ = 2x and v’ = eˣ, giving dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x).
对于 y = x² eˣ，设 u = x²，v = eˣ。则 u’ = 2x，v’ = eˣ，得到 dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x)。
6. Quotient Rule | 商法则
For y = u/v, the quotient rule is dy/dx = (vu’ − uv’)/v². It is needed for rational functions such as y = (x² + 1)/(x − 3). Here u = x² + 1, v = x − 3, so u’ = 2x, v’ = 1, giving dy/dx = [(x − 3)(2x) − (x² + 1)(1)]/(x − 3)² = (x² − 6x − 1)/(x − 3)².
对于 y = u/v，商法则为 dy/dx = (vu’ − uv’)/v²。它适用于有理函数，例如 y = (x² + 1)/(x − 3)。这里 u = x² + 1，v = x − 3，因此 u’ = 2x，v’ = 1，得到 dy/dx = [(x − 3)(2x) − (x² + 1)(1)]/(x − 3)² = (x² − 6x − 1)/(x − 3)²。
Remember that the order in the numerator matters: it is v times u’ first, then u times v’. Do not forget to square the denominator.
记住分子中的顺序很重要：先写 v 乘以 u’，再写 u 乘以 v’。不要忘记将分母平方。
7. Tangents and Normals | 切线与法线
The derivative gives the gradient m of the tangent at a point (x₁, y₁). The tangent equation is y − y₁ = m(x − x₁). The normal is perpendicular to the tangent, so its gradient is −1/m, and its equation is y − y₁ = (−1/m)(x − x₁).
导数给出点 (x₁, y₁) 处切线的斜率 m。切线方程为 y − y₁ = m(x − x₁)。法线垂直于切线，因此其斜率为 −1/m，其方程为 y − y₁ = (−1/m)(x − x₁)。
For example, if y = x² and m = 4 at x = 2, the tangent at (2, 4) is y − 4 = 4(x − 2). The normal has gradient −1/4, so its equation is y − 4 = −(1/4)(x − 2).
例如，若 y = x² 且在 x = 2 处 m = 4，则 (2, 4) 处的切线为 y − 4 = 4(x − 2)。法线的斜率为 −1/4，因此其方程为 y − 4 = −(1/4)(x − 2)。
8. Stationary Points | 驻点
Stationary points occur where dy/dx = 0. To classify them, use the second derivative d²y/dx². If d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum; if d²y/dx² = 0, further investigation is needed, such as checking the sign of dy/dx on either side.
驻点出现在 dy/dx = 0 处。使用二阶导数 d²y/dx² 来判定其性质。如果 d²y/dx² > 0，该点是局部极小值；如果 d²y/dx² < 0，它是局部极大值；如果 d²y/dx² = 0，需要进一步研究，例如检查两侧 dy/dx 的符号。
For y = x³ − 3x, dy/dx = 3x² − 3 = 3(x² − 1), so stationary points are at x = 1 and x = −1. Since d²y/dx² = 6x, at x = 1 it is positive, giving a minimum, and at x = −1 it is negative, giving a maximum.
对于 y = x³ − 3x，dy/dx = 3x² − 3 = 3(x² − 1)，因此驻点在 x = 1 和 x = −1 处。由于 d²y/dx² = 6x，在 x = 1 处为正，给出极小值；在 x = −1 处为负，给出极大值。
9. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative measures the rate of change of the gradient. A positive second derivative means the curve is convex (curving upward), while a negative second derivative means it is concave (curving downward). Points of inflection occur where the concavity changes; they may be stationary or non-stationary.
二阶导数衡量斜率的变化率。正的二阶导数表示曲线是下凸的（向上弯曲），而负的二阶导数表示曲线是上凸的（向下弯曲）。拐点出现在凹凸性改变处；它们可以是驻点，也可以是非驻点。
At a non-stationary point of inflection, dy/dx is not necessarily zero, but d²y/dx² = 0 or is undefined, and the sign of d²y/dx² changes on either side of the point.
在非驻点拐点处，dy/dx 不一定为零，但 d²y/dx² = 0 或不存在，并且 d²y/dx² 的符号在该点两侧发生变化。
10. Parametric Differentiation | 参数方程求导
When x and y are given in terms of a parameter t, dy/dx = (dy/dt)/(dx/dt). For example, if x = t² + 1 and y = 2t³, then dx/dt = 2t and dy/dt = 6t², so dy/dx = 3t.
当 x 和 y 用参数 t 表示时，dy/dx = (dy/dt)/(dx/dt)。例如，若 x = t² + 1，y = 2t³，则 dx/dt = 2t，dy/dt = 6t²，所以 dy/dx = 3t。
The second derivative requires d²y/dx² = d/dt(dy/dx) / (dx/dt). In this example, d/dt(3t) = 3 and dx/dt = 2t, so d²y/dx² = 3/(2t).
二阶导数需要 d²y/dx² = d/dt(dy/dx) / (dx/dt)。在这个例子中，d/dt(3t) = 3，dx/dt = 2t，所以 d²y/dx² = 3/(2t)。
11. Connected Rates of Change | 相关变化率
In many modelling problems, two or more quantities change with time but are related by a geometric or physical equation. If y = f(x) and x = g(t), then dy/dt = (dy/dx)(dx/dt). This is used for expanding circles, filling containers, and moving ladders.
在许多建模问题中，两个或多个量随时间变化，但通过几何或物理方程相互关联。如果
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