📚 Mastering Differentiation: Rules, Techniques and Applications for Edexcel A-Level Maths | Edexcel A-Level 数学微分核心考点与技巧
Differentiation is one of the central pillars of Edexcel A-Level Mathematics. It allows you to measure how one quantity changes with respect to another, and it underpins topics from graph sketching to optimisation and kinematics. This revision guide covers the essential rules, techniques and applications you need to master for the Pure Mathematics papers.
微分是 Edexcel A-Level 数学的核心支柱之一。它帮助你衡量一个量相对于另一个量的变化率，并支撑从曲线草图、优化到运动学的多个主题。本复习指南涵盖纯数学试卷中必须掌握的基本法则、技巧与应用。
1. Definition and First Principles | 定义与第一性原理
The derivative f'(x) is defined as the limit of the average rate of change as h tends to zero. This first-principles definition is examinable and helps you understand what a derivative actually measures: the instantaneous rate of change of f(x) with respect to x.
导数 f'(x) 被定义为当 h 趋于零时平均变化率的极限。这个第一性原理定义是考试可能考查的内容，并能帮助你理解导数实际衡量的是什么：f(x) 相对于 x 的瞬时变化率。
f ‘(x) = limh→0 [f(x + h) − f(x)] / h
To differentiate from first principles, substitute f(x+h) into the expression, simplify the numerator, cancel h where possible, then take the limit as h approaches 0. For example, for f(x) = x², the expansion (x+h)² = x² + 2xh + h² leads to f'(x) = 2x after cancelling h.
要从第一性原理求导，需要将 f(x+h) 代入表达式，化简分子，尽可能约去 h，然后令 h 趋于 0 取极限。例如，对于 f(x) = x²，展开 (x+h)² = x² + 2xh + h²，约去 h 后得到 f'(x) = 2x。
2. Power Rule and Basic Derivatives | 幂法则与基本导数
The power rule states that if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This rule works for any real constant n, including negative and fractional powers such as x⁻² and √x. You should not assume n is a positive integer.
幂法则表明，如果 f(x) = xⁿ，那么 f'(x) = n xⁿ⁻¹。该法则适用于任何实数常量 n，包括负指数和分数指数，例如 x⁻² 和 √x。不要默认 n 一定是正整数。
d/dx (xⁿ) = n xⁿ⁻¹
Before differentiating, you should often rewrite expressions using index laws. For example, 1/x² becomes x⁻², and √x becomes x^(1/2). Then apply the power rule term by term. For y = 3x⁴ − 5/x + 2√x, rewrite as y = 3x⁴ − 5x⁻¹ + 2x^(1/2), giving dy/dx = 12x³ + 5x⁻² + x^(−1/2).
在求导之前，通常应先用指数法则重写表达式。例如，1/x² 写成 x⁻²，√x 写成 x^(1/2)。然后逐项应用幂法则。对于 y = 3x⁴ − 5/x + 2√x，先重写为 y = 3x⁴ − 5x⁻¹ + 2x^(1/2)，得到 dy/dx = 12x³ + 5x⁻² + x^(−1/2)。
3. Product, Quotient and Chain Rules | 乘积、商和链式法则
For the product y = u v, the product rule is dy/dx = u’v + uv’. For the quotient y = u/v, the quotient rule is dy/dx = (u’v − uv’) / v². Both rules require you to identify u and v clearly and compute u’ and v’ before substituting into the formula.
对于乘积 y = u v，乘积法则是 dy/dx = u’v + uv’。对于商 y = u/v，商法则是 dy/dx = (u’v − uv’) / v²。两种法则都要求先明确 u 和 v，并计算 u’ 和 v’，再代入公式。
dy/dx = u’v + uv’ and dy/dx = (u’v − uv’) / v²
The chain rule is used for composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In Leibniz notation, dy/dx = (dy/du)(du/dx). This is the most frequently used differentiation technique in A-Level questions, so practise differentiating powers, exponentials, logs and trigonometric functions of linear or mixed expressions.
链式法则用于复合函数：若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。用莱布尼茨记号表示为 dy/dx = (dy/du)(du/dx)。这是 A-Level 题目中最常用的求导技巧，因此要练习对线性或混合表达式进行幂、指数、对数及三角函数的复合求导。
4. Differentiating Exponential and Logarithmic Functions | 指数与对数函数求导
The natural exponential function has the special property d/dx(eˣ) = eˣ. For a general exponential aˣ, use d/dx(aˣ) = aˣ ln a. When the exponent is a function of x, apply the chain rule: d/dx(e^(kx)) = k e^(kx).
自然指数函数具有特殊性质 d/dx(eˣ) = eˣ。对于一般指数函数 aˣ，使用 d/dx(aˣ) = aˣ ln a。当指数是 x 的函数时，应用链式法则：d/dx(e^(kx)) = k e^(kx)。
d/dx (eˣ) = eˣ and d/dx (aˣ) = aˣ ln a
The derivative of ln x is 1/x for x > 0. For ln(f(x)), apply the chain rule to obtain f'(x)/f(x). This is particularly useful for differentiating expressions such as y = ln(3x² + 1), which gives dy/dx = 6x / (3x² + 1).
ln x 的导数是 1/x（x > 0）。对于 ln(f(x))，应用链式法则得到 f'(x)/f(x)。这一结论在求导如 y = ln(3x² + 1) 时特别有用，其结果为 dy/dx = 6x / (3x² + 1)。
5. Differentiating Trigonometric Functions | 三角函数求导
The standard trigonometric derivatives are d/dx(sin x) = cos x, d/dx(cos x) = −sin x, and d/dx(tan x) = sec² x. For composite angles such as sin(kx) or cos(kx), multiply by the derivative of the angle using the chain rule: d/dx(sin kx) = k cos kx.
标准三角函数的导数为 d/dx(sin x) = cos x，d/dx(cos x) = −sin x，d/dx(tan x) = sec² x。对于复合角如 sin(kx) 或 cos(kx)，使用链式法则乘以角的导数：d/dx(sin kx) = k cos kx。
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x
These results assume x is measured in radians. Always switch your calculator to radian mode when differentiating or evaluating trigonometric derivatives, unless a question explicitly states that angles are in degrees. Using degrees will produce incorrect gradient values.
这些结果假设 x 以弧度为单位。在求导或计算三角函数导数时，除非题目明确使用角度制，否则始终将计算器切换到弧度模式。使用角度制会得到错误的梯度值。
6. Implicit Differentiation | 隐函数求导
Implicit differentiation is used when y is not expressed explicitly as a function of x. Differentiate both sides of an equation with respect to x, treating y as a function of x and multiplying by dy/dx whenever you differentiate a term involving y.
当 y 未显式表示为 x 的函数时，使用隐函数求导。对等式两边关于 x 求导，将 y 视为 x 的函数，并在对含有 y 的项求导时乘以 dy/dx。
d/dx (y²) = 2y dy/dx
This technique is particularly useful for circles, ellipses and other curves like x² + y² = r², and it allows you to find gradients at points without solving for y. After differentiating, collect all dy/dx terms on one side and factorise to solve for dy/dx explicitly.
该技巧特别适用于圆、椭圆以及其他曲线，如 x² + y² = r²，并且无需解出 y 即可求出某点处的梯度。求导后，将所有包含 dy/dx 的项移到一边并因式分解，就能显式解出 dy/dx。
7. Parametric Differentiation | 参数方程求导
When x and y are both given in terms of a parameter t, the derivative dy/dx is found using dy/dx = (dy/dt) / (dx/dt), provided dx/dt is not zero. This formula follows directly from the chain rule and is essential for curves defined parametrically.
当 x 和 y 都用参数 t 表示时，导数 dy/dx 通过 dy/dx = (dy/dt) / (dx/dt) 求出，前提是 dx/dt 不为零。该公式直接来自链式法则，是处理参数定义曲线的基础。
dy/dx = (dy/dt) / (dx/dt)
Parametric differentiation is common in questions involving curves such as x = f(t), y = g(t). You can also use it to find tangents and stationary points when the Cartesian equation is difficult to obtain. For stationary points, set dy/dx = 0, which normally means dy/dt = 0 while dx/dt ≠ 0.
参数方程求导常见于涉及 x = f(t)、y = g(t) 等曲线的题目中。当笛卡尔方程难以获得时，也可以用此方法求切线和驻点。求驻点时，令 dy/dx = 0，通常意味着 dy/dt = 0 且 dx/dt ≠ 0。
8. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative f”(x) or d²y/dx² measures the rate of change of the gradient. If f”(x) > 0 on an interval, the curve is convex (or concave up); if f”(x) < 0, it is concave (or concave down). This information is often linked to the shape of a graph.
二阶导数 f”(x) 或 d²y/dx² 衡量梯度的变化率。如果在某区间内 f”(x) > 0，曲线是凸的（向上凹）；如果 f”(x) < 0，曲线是凹的（向下凹）。这一信息通常与图形形状相关。
f ”(x) = d²y/dx²
Points where the concavity changes are called points of inflection. At such points f”(x) may be zero, but you must check the sign change to confirm the nature. A point where f”(x) = 0 is not automatically a point of inflection; the second derivative must change sign around that point.
凹性发生变化的点称为拐点。在这些点处 f”(x) 可能为零，但必须检查符号变化以确认其性质。f”(x) = 0 的点不一定就是拐点；二阶导数必须在该点附近改变符号。
9. Stationary Points and Their Classification | 驻点及其分类
Stationary points occur where f'(x) = 0. To classify them, find the second derivative: if f”(x) > 0 the point is a local minimum; if f”(x) < 0 it is a local maximum. This second derivative test is quick and often sufficient in Edexcel questions.
驻点出现在 f'(x) = 0 处。要对其分类，可求二阶导数：若 f”(x) > 0，该点为局部极小值；若 f”(x) < 0，该点为局部极大值。这种二阶导数检验快速且足以应对 Edexcel 题目。
f ”(x) > 0 ⇒ minimum  
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