📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | Edexcel A-Level 纯数学微分精讲
Differentiation is a central topic in Edexcel A-Level Pure Mathematics. It allows us to calculate exact rates of change, analyse the behaviour of graphs, and solve real-world optimisation problems. In Papers 1 and 2, you can expect differentiation to appear in a wide range of question styles, from simple derivative calculations to connected rates of change and implicit equations.
微分是 Edexcel A-Level 纯数学的核心主题。它使我们能够计算精确的变化率、分析图形的行为，并解决现实世界中的最优化问题。在 Paper 1 和 Paper 2 中，微分几乎会以各种题型出现，从简单的求导计算到相关变化率和隐函数方程。
1. What Differentiation Measures | 微分的意义
Differentiation gives the gradient of a curve at a point. For a function y = f(x), the derivative dy/dx is the rate at which y changes with respect to x. Geometrically, dy/dx at x = a is the gradient of the tangent to the curve at the point (a, f(a)). This concept underpins tangent and normal equations, velocity from displacement, and marginal cost from total cost.
微分给出曲线在某一点的梯度。对于函数 y = f(x)，导数 dy/dx 表示 y 随 x 变化的瞬时变化率。从几何上看，dy/dx 在 x = a 处的值就是曲线在点 (a, f(a)) 处切线的斜率。这一概念是求解切线法线方程、由位移求速度以及由总成本求边际成本的基础。
2. First Principles Definition | 第一性原理定义
The derivative of f(x) from first principles is defined as:
f'(x) = limₕ→0 [f(x + h) − f(x)] / h
This limit represents the gradient of the chord between two nearby points on the curve as the distance h tends to zero. Examiners often ask for a proof or a small application of this definition for functions such as x² or 1/x. You must write the limit notation clearly and show each algebraic step.
函数 f(x) 的导数从第一性原理出发定义为：
f'(x) = limₕ→0 [f(x + h) − f(x)] / h
该极限表示当两点间距离 h 趋近于零时，曲线上两个邻近点之间弦的斜率。考官经常要求用此定义证明或简单应用到 x² 或 1/x 等函数。你必须清楚地写出极限符号，并展示每一步代数变形。
3. Standard Derivatives and the Power Rule | 标准导数与幂法则
For any real constant n, the power rule gives:
If f(x) = xⁿ, then f'(x) = nxⁿ⁻¹
You should memorise standard results for trigonometric, exponential and logarithmic functions. For Edexcel A-Level, key derivatives include:
| Function f(x) | Derivative f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| eˣ | eˣ |
| ln x | 1/x |
对于任意实数常数 n，幂函数求导法则给出：
如果 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹
你需要熟记三角函数、指数函数和对数函数的标准导数。上表列出了 Edexcel A-Level 必须掌握的关键结果。考试中很多综合题都依赖这些基础导数，若记忆不牢会直接影响后续计算。
4. Product, Quotient and Chain Rules | 乘积法则、商法则与链式法则
When a function is a product of two functions, use the product rule. If y = u(x)v(x), then dy/dx = u’v + uv’. For a quotient y = u/v, use the quotient rule: dy/dx = (vu’ − uv’) / v². The chain rule handles composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x).
当一个函数是两个函数的乘积时，使用乘积法则。如果 y = u(x)v(x)，则 dy/dx = u’v + uv’。对于商函数 y = u/v，使用商法则：dy/dx = (vu’ − uv’) / v²。链式法则处理复合函数：如果 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。
Always identify u and v clearly in working. For example, y = x² sin x has u = x² and v = sin x, giving dy/dx = 2x sin x + x² cos x. Many exam errors come from misapplying the quotient rule numerator order, so write it as ‘bottom times derivative of top minus top times derivative of bottom, all over bottom squared’.
解题时务必清楚地标出 u 和 v。例如 y = x² sin x 中令 u = x²，v = sin x，则 dy/dx = 2x sin x + x² cos x。很多考试失分来自商法则分子顺序写反，所以建议记住口诀 ‘下面乘上面导数减去上面乘下面导数，再除以下面平方’。
5. Implicit Differentiation | 隐函数微分
Implicit differentiation is used when y is not written explicitly as a function of x. Differentiate both sides term by term with respect to x, treating y as a function of x. Every time you differentiate a term involving y, multiply by dy/dx using the chain rule, for example d/dx (y²) = 2y dy/dx.
隐函数微分用于 y 没有显式写成 x 的函数的情况。对等式两边逐项关于 x 求导，并将 y 视为 x 的函数。每次对含 y 的项求导时，都要用链式法则乘以 dy/dx，例如 d/dx (y²) = 2y dy/dx。
A typical Edexcel question might be: x² + y² = 25. Differentiating gives 2x + 2y dy/dx = 0, so dy/dx = −x/y. Then you can substitute coordinates to find the gradient at a given point.
Edexcel 的典型题目可能是：x² + y² = 25。求导得 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。然后代入坐标即可求出给定点处的梯度。
6. Parametric Differentiation | 参数方程微分
When a curve is defined parametrically by x = x(t) and y = y(t), the chain rule gives dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero. This allows you to find tangent and normal equations without eliminating the parameter.
当曲线由参数方程 x = x(t) 和 y = y(t) 定义时，由链式法则得 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt 不为零。这样无需消去参数即可求出切线和法线方程。
For example, x = t² + 1 and y = 2t. Then dx/dt = 2t and dy/dt = 2, so dy/dx = 2 / (2t) = 1/t. At t = 2, the gradient is 1/2 and the point is (5, 4).
例如 x = t² + 1，y = 2t。则 dx/dt = 2t，dy/dt = 2，所以 dy/dx = 2 / (2t) = 1/t。当 t = 2 时，梯度为 1/2，对应点为 (5, 4)。
7. Second Order Derivatives | 二阶导数
The second derivative d²y/dx² is the rate of change of the first derivative. It is found by differentiating dy/dx with respect to x. It tells us about the concavity of a curve and helps classify stationary points.
二阶导数 d²y/dx² 是一阶导数的变化率。它是将 dy/dx 再对 x 求导得到的。它告诉我们曲线的凹凸性，并帮助判断驻点类型。
For parametric equations, the second derivative is not simply (d²y/dt²)/(d²x/dt²). You must use d²y/dx² = (d/dx)(dy/dx) = (d/dt)(dy/dx) / (dx/dt). This is a common exam trap.
对于参数方程，二阶导数并不是简单地等于 (d²y/dt²)/(d²x/dt²)。你必须使用 d²y/dx² = (d/dx)(dy/dx) = (d/dt)(dy/dx) / (dx/dt)。这是一个常见的考试陷阱。
8. Stationary Points and Classification | 驻点及其分类
A stationary point occurs where dy/dx = 0. There are three types: local maximum, local minimum, and point of inflection. To classify a stationary point, you can use the sign of dy/dx either side of the point, or the
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