📚 Mastering Differentiation for Edexcel A-Level Pure Mathematics | 掌握 Edexcel A-Level 纯数学微分
In Edexcel A-Level Mathematics, differentiation is one of the most heavily examined topics in Pure Mathematics. It underpins rates of change, gradient calculations, curve sketching, optimisation and many real-world modelling problems. This revision guide breaks down the key techniques, notation and common question types you need to master.
在 Edexcel A-Level 数学中，微分是纯数学部分考查频率最高的主题之一。它支撑着变化率、斜率计算、曲线作图、优化问题以及许多现实建模问题。本复习指南将拆解你需要掌握的关键技巧、符号和常见题型。
1. Why Differentiation Matters in Edexcel A-Level | 微分在 Edexcel A-Level 中的重要性
Differentiation appears throughout the Edexcel Pure Mathematics specification, especially in Pure 1, Pure 2 and the Year 13 pure papers. It is not an isolated technique: it connects algebra, trigonometry, exponentials, logarithms, coordinate geometry and modelling.
微分贯穿 Edexcel 纯数学考纲，尤其是 Pure 1、Pure 2 和 Year 13 纯数试卷。它不是一个孤立技巧：它把代数、三角、指数、对数、坐标几何和建模连接起来。
Typical exam questions ask you to find a gradient, equation of a tangent or normal, stationary point, rate of change, or maximum/minimum value in a modelling context. Later topics such as integration, differential equations and numerical methods also depend on confident differentiation.
典型考题要求你求斜率、切线或法线方程、驻点、变化率，或在建模背景中求最大/最小值。后续的积分、微分方程和数值方法等主题也依赖熟练的微分能力。
Marks are often awarded for correct notation, clear substitution and valid reasoning, not only for the final answer. Therefore, you should practise writing every step as if the examiner cannot read your mind.
分数往往不仅给最终答案，也给正确的符号、清晰的代入和有效推理。因此，你应练习把每一步都写清楚，就像考官无法读你的心一样。
2. The Gradient Function and First Principles | 梯度函数与第一原理
asures the gradient of the tangent to the curve at any point. It is defined as the limit of the average rate of change as the interval h tends to zero.
对于曲线 y = f(x)，导数衡量曲线上任意一点处切线的斜率。它被定义为当区间 h 趋于零时平均变化率的极限。
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
Alternative notation includes dy/dx, d/dx [f(x)] and, when the variable is time, f'(t) or dx/dt. The notation you choose should match the question and remain consistent in your working.
其他记号包括 dy/dx、d/dx [f(x)]，当变量为时间时也可写 f'(t) 或 dx/dt。你选择的记号应与题目一致，并在解题过程中保持一致。
fter dividing by h, the expression becomes 2x + h, and taking the limit as h → 0 gives 2x.
例如，当 f(x) = x² 时，展开 (x + h)² − x² 得到 2xh + h²。除以 h 后表达式变为 2x + h，令 h → 0 取极限得到 2x。
First-principles questions are usually clearly signposted by phrases such as 'from first principles' or 'using the definition of the derivative'. Do not use the standard rule in these questions unless you are told to do so.
第一原理题通常由 'from first principles' 或 'using the definition of the derivative' 等措辞明确提示。除非题目允许，否则在这类题中不要直接使用标准求导法则。
3. Standard Derivatives and Basic Rules | 标准导数与基本法则
You must know the standard derivatives of powers, trigonometric functions, exponentials and logarithms. The constant multiple rule and sum rule allow you to differentiate term by term.
你必须掌握幂函数、三角函数、指数函数和对数函数的标准导数。常数倍法则和加法法则使你能够逐项求导。
d/dx (xⁿ) = n xⁿ⁻¹ for any real n, including negative and fractional powers.
d/dx (xⁿ) = n xⁿ⁻¹，对任意实数 n 成立，包括负指数和分数指数。
d/dx (sin x) = cos x, d/dx (cos x) = −sin x, d/dx (tan x) = sec² x.
d/dx (sin x) = cos x，d/dx (cos x) = −sin x，d/dx (tan x) = sec² x。
d/dx (eˣ) = eˣ and d/dx (ln x) = 1/x for x > 0.
d/dx (eˣ) = eˣ，且当 x > 0 时 d/dx (ln x) = 1/x。
If y = a f(x) + b g(x), then dy/dx = a f'(x) + b g'(x).
若 y = a f(x) + b g(x)，则 dy/dx = a f'(x) + b g'(x)。
x² − 2 cos x. The constant 5 differentiates to zero because a horizontal line has zero gradient.
例如，d/dx (4x³ − 2 sin x + 5) = 12x² − 2 cos x。常数 5 的导数为零，因为水平线的斜率为零。
Remember that angles in trigonometric differentiation must be measured in radians. If a question uses degrees, convert to radians before differentiating unless the question explicitly asks for a degrees-based result.
记住，三角函数微分中的角度必须以弧度为单位。若题目使用角度制，除非明确要求角度制结果，否则应先转换为弧度再求导。
4. The Chain Rule | 链式法则
e function is nested inside another. If y = f(u) and u = g(x), then dy/dx = dy/du × du/dx.
链式法则用于一个函数嵌套在另一个函数内部的情况。如果 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。
In words, differentiate the outer function, keep the inner expression unchanged, and then multiply by the derivative of the inner expression.
换句话说，先对外层函数求导，保持内层表达式不变，然后乘以内层表达式的导数。
= u⁴. Then dy/du = 4u³ and du/dx = 6x. Therefore dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.
例如：y = (3x² + 5)⁴。令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x。因此 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。
Further examples include d/dx [sin(2x + 1)] = 2 cos(2x + 1), d/dx [e3x²] = 6x e3x², and d/dx [ln(5x + 2)] = 5/(5x + 2).
更多例子包括 d/dx [sin(2x + 1)] = 2 cos(2x + 1)，d/dx [e3x²] = 6x e3x²，以及 d/dx [ln(5x + 2)] = 5/(5x + 2)。
The most common mistake is forgetting to multiply by the derivative of the inner function. Always ask: what is inside the bracket, exponent, sine, cosine, exponential or logarithm?
最常见的错误是忘记乘以内层函数的导数。始终问自己：括号、指数、正弦、余弦、指数函数或对数里面的内容是什么？
5. The Product Rule and Quotient Rule | 乘积法则与商法则
When two expressions are multiplied, use the product rule: if y = uv, then dy/dx = u'v + uv' n one expression is divided by another, use the quotient rule: if y = u/v, then dy/dx = (u'v − uv') / v².
当两个表达式相乘时，使用乘积法则：若 y = uv，则 dy/dx = u'v + uv'。当一个表达式除以另一个表达式时，使用商法则：若 y = u/v，则 dy/dx = (u'v − uv') / v²。
Example with the product rule: y = x² sin x. Let u = x² and v = sin x. Then u' = 2x and v' = cos x, so dy/dx = 2x sin x + x² cos x.
乘积法则示例：y = x² sin x。令 u = x²，v = sin x。则 u' = 2x，v' = cos x，所以 dy/dx = 2x sin x + x² cos x。
Example with the quotient rule: y = (2x + 1)/(x − 3). Here u = 2x + 1 and v = x − 3, so u' = 2 and v' dy/dx = [2(x − 3) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1) / (x − 3)² = −7/(x − 3)².
商法则示例：y = (2x + 1)/(x − 3)。这里 u = 2x + 1，v = x − 3，所以 u' = 2，v' = 1。于是 dy/dx = [2(x − 3) − (2x + 1)(1)] / (x − 3)² = (2x − 6 − 2x − 1) / (x − 3)² = −7/(x − 3)²。
Choose the product rule when the expression is a product, and the quotient rule when it is a single fraction. Sometimes rewriting a quotient as a product with a negative power is faster, especially for simple powers.
表达式是乘积时选择乘积法则，是单个分式时选择商法则。有时把商改写成带负指数的乘积会更快，尤其是简单幂函数时。
6. Tangents, Normals and Rates of Change | 切线、法线与变化率
The gradient of the tangent to y = f(x) at x = a is f'. The tangent line through the point (x₁, y₁) has equation y − y₁ = m(x − x₁), where m = f'(x₁).
曲线 y = f(x) 在 x = a 处切线的斜率是 f'(a)。过点 (x₁, y₁) 的切线方程为 y − y₁ = m(x − x₁)，其中 m = f'(x₁)。
The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0. The normal equation uses the same point but the negative reciprocal gradient.
法线与切线垂直，因此其斜率为 −1/m，前提是 m ≠ 0。法线方程使用同一点，但斜率取负倒数。
− 3 = 12 − 3 = 9. The tangent is y − 2 = 9(x − 2), and the normal is y − 2 = −1/9 (x − 2).
示例：y = x³ − 3x。在 x = 2 时，y = 8 − 6 = 2，dy/dx = 3x² − 3 = 12 − 3 = 9。切线为 y − 2 = 9(x − 2)，法线为 y − 2 = −1/9 (x − 2)。
x and x depends on t, then dy/dt = dy/dx × dx/dt. This is the chain rule in applied form.
变化率用同样的思想处理时间或其他变量。若 y 依赖于 x，x 依赖于 t，则 dy/dt = dy/dx × dx/dt。这是链式法则的应用形式。
d acceleration a = dv/dt = d²s/dt². Units must be stated correctly in modelling questions.
在运动学中，若位移 s 表示为时间 t 的函数，则速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。建模题中必须正确写出单位。
7. Stationary Points, Second Derivative and Curve Sketching | 驻点、二阶导数与曲线作图
< 0, it is a local maximum; if d²y/dx² = 0, the test is inconclusive and you should check the sign of dy/dx on both sides.
驻点出现在 dy/dx = 0 处。要判断其类型，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；若 d²y/dx² = 0，该检验不确定，应检查 dy/dx 在两侧的符号。
= 3x² − 3 = 3(x² − 1), so stationary points occur at x = 1 and x = −1. Also d²y/dx² = 6x.
示例：y = x³ − 3x。则 dy/dx = 3x² − 3 = 3(x² − 1)，所以驻点在 x = 1 和 x = −1 处。又 d²y/dx² = 6x。
t (−1, 2). The curve also has a point of inflection at x = 0, where the concavity changes.
在 x = 1 处，d²y/dx² = 6 > 0，所以 (1, −2) 为局部极小值点。在 x = −1 处，d²y/dx² = −6 < 0，所以 (−1, 2) 为局部极大值点。曲线在 x = 0 处还有拐点，凹凸性在此改变。
For curve sketching, combine the y-intercept, roots, stationary points, points of inflection, asymptotic behaviour and the sign of the gradient in each interval. A sketch should show the shape clearly and label any key coordinates.
曲线作图时，结合 y 截距、根、驻点、拐点、渐近行为以及各区间内梯度的符号。草图应清晰显示形状并标出关键坐标。
8. Optimisation and Modelling Problems | 优化与建模问题
Optimisation problems ask for a maximum or minimum value in a real context, such as area, volume, cost or profit. The standard method is to form a single-variable function, differentiate it, set the derivative to zero and justify the nature of the stationary point.
优化问题要求在真实情境中求最大值或最小值，例如面积、体积、成本或利润。标准方法是建立一个单变量函数，求导，令导数为零，并说明驻点的性质。
the second derivative or a sign test; check the domain and answer the question in context.
步骤：定义变量和单位；写出约束条件；建立目标函数；求导；解 dy/dx = 0；用二阶导数或符号检验判断类型；检查定义域并结合情境回答问题。
idth be y cm. Then 2x + 2y = 40, so y = 20 − x. The area is A = xy = x(20 − x) = 20x − x².
示例：一个矩形周长为 40 cm。设长为 x cm，宽为 y cm。则 2x + 2y = 40，所以 y = 20 − x。面积为 A = xy = x(20 − x) = 20x − x²。
maximum area is 100 cm². The shape is a square, which is a useful result to remember for fixed-perimeter rectangles.
求导：dA/dx = 20 − 2x。令 dA/dx = 0 得 x = 10，所以 y = 10。由于 d²A/dx² = −2 < 0，这是最大值。最大面积为 100 cm²。此时形状为正方形，这是固定周长矩形的有用结论。
Always re-read the question to ensure you give the quantity requested. Some questions ask for the dimensions, others for the maximum value, and some for both. Include correct units in your final answer.
务必重读题目，确保给出所要求的量。有些题要求尺寸，有些要求最大值，有些两者都要。最终答案中要包含正确单位。
9. Implicit and Parametric Differentiation | 隐函数与参数方程微分
Implicit differentiation is used when y is not given explicitly as a function of x. Differentiate both sides with respect to x, remembering that every derivative of y produces a factor dy/dx.
当 y 未显式表示为 x 的函数时，使用隐函数微分。对两边关于 x 求导，记住每一项对 y 的导数都会产生因子 dy/dx。
= 0, so dy/dx = −x/y, provided y ≠ 0. This is the gradient of a circle at a general point.
示例：x² + y² = 25。求导得 2x + 2y dy/dx = 0，所以当 y ≠ 0 时 dy/dx = −x/y。这是圆在一般点处的斜率。
x = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This is a direct application of the chain rule.
对于参数方程，若 x = x(t)，y = y(t)，则当 dx/dt ≠ 0 时，dy/dx = (dy/dt) / (dx/dt)。这是链式法则的直接应用。
/2 for t ≠ 0. If needed, substitute the parameter value to find the gradient at a specific point.
示例：x = t²，y = t³。则 dx/dt = 2t，dy/dt = 3t²，所以当 t ≠ 0 时 dy/dx = 3t²/(2t) = 3t/2。如有需要，代入参数值以求出特定点处的斜率。
With implicit differentiation, product terms require the product rule. For example, differentiating xy with respect to x gives x dy/dx + y, not just x dy/dx.
隐函数微分中，乘积项需要乘积法则。例如，对 xy 关于 x 求导得到 x dy/dx + y，而不只是 x dy/dx。
10. Connected Rates of Change | 连通变化率
Connected rates of change problems link two or more quantities that change with time. The chain rule is used to connect the rate you know to the rate you need.
连通变化率问题把两个或多个随时间变化的量联系起来。使用链式法则把你已知的变化率与你需要的变化率连接起来。
d so that its volume increases at 100 cm³/s. Given V = 4/3 π r³, find dr/dt when r = 5 cm.
示例：一个球形气球充气，体积以 100 cm³/s 增加。已知 V = 4/3 π r³，求 r = 5 cm 时的 dr/dt。
refore 100 = 4π(5)² × dr/dt = 100π × dr/dt, so dr/dt = 100/(100π) = 1/π cm/s ≈ 0.318 cm/s.
对体积关于半径求导：dV/dr = 4πr²。由链式法则，dV/dt = dV/dr × dr/dt。因此 100 = 4π(5)² × dr/dt = 100
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