📚 A-Level Edexcel Pure Maths: Differentiation Essentials | Edexcel A-Level 纯数学：微分核心要点
Differentiation is one of the most heavily examined topics in Edexcel A-Level Pure Mathematics. It is used to find tangents, normals, stationary points, increasing and decreasing intervals, and to solve optimisation and modelling problems. Mastering the basic rules and common exam techniques is essential for success in both AS and A2 papers.
微分是Edexcel A-Level纯数学中考查频率最高的主题之一。它用于求解切线、法线、驻点、递增与递减区间，以及解决优化和建模问题。掌握基本法则和常见考试技巧，对AS和A2试卷的成功至关重要。
1. Gradient Function and First Principles | 梯度函数与第一性原理
The derivative of a function f(x) is defined by the limit of the average rate of change as h approaches zero. This is known as differentiation from first principles.
函数f(x)的导数定义为当h趋近于零时平均变化率的极限。这称为从第一性原理出发的微分。
f'(x) = lim (h → 0) [f(x+h) – f(x)] / h
If this limit exists at a point x, then f is differentiable at x. The derivative is also written as dy/dx when y = f(x). In Edexcel exam questions, you may be asked to prove the derivative of a simple function such as x² using this definition.
如果该极限在点x处存在，则函数f在x处可导。当y = f(x)时，导数也写作dy/dx。在Edexcel考试题中，可能会要求你用这个定义证明x²等简单函数的导数。
Example: For f(x) = x², expand f(x+h) – f(x) = 2xh + h², divide by h to get 2x + h, and let h → 0 to obtain f'(x) = 2x.
例如：对于f(x) = x²，展开f(x+h) – f(x) = 2xh + h²，除以h得到2x + h，令h → 0，即得f'(x) = 2x。
2. Power Rule and Polynomials | 幂函数法则与多项式
The most frequently used rule in A-Level differentiation is the power rule. It allows you to differentiate any single power of x quickly.
A-Level微分中最常用的法则是幂函数法则。它可以快速对任意单个x的幂求导。
d/dx (xⁿ) = nxⁿ⁻¹
This rule works for any real constant n, including negative and fractional powers. For example, x⁻² differentiates to -2x⁻³, and x^½ differentiates to ½x⁻½. You should be comfortable rewriting roots and reciprocals before applying the rule.
该法则适用于任意实数常量n，包括负指数和分数指数。例如，x⁻²的导数为-2x⁻³，x^½的导数为½x⁻½。你应当先熟练地将根式和倒数改写为幂的形式，再应用该法则。
Example: y = 3x⁵ – 2x³ + 7x – 4 gives dy/dx = 15x⁴ – 6x² + 7.
例如：y = 3x⁵ – 2x³ + 7x – 4，则dy/dx = 15x⁴ – 6x² + 7。
3. Constant Multiple and Sum Rules | 常数倍法则与和法则
Differentiation is linear, so you can differentiate term by term. Constant factors remain unchanged, and the derivative of a sum is the sum of the derivatives.
微分具有线性性质，因此可以逐项求导。常数因子保持不变，和的导数等于导数之和。
d/dx [a f(x) + b g(x)] = a f'(x) + b g'(x)
Constants on their own differentiate to zero. This is a common source of marks in exam questions: after differentiating a polynomial, the constant term disappears.
单独的常数求导为零。这是考试题中常见的得分点：对多项式求导后，常数项会消失。
Example: y = 4x³ – 2x + 7 gives dy/dx = 12x² – 2 because the derivative of 7 is zero.
例如：y = 4x³ – 2x + 7，得到dy/dx = 12x² – 2，因为7的导数为零。
4. Tangents and Normals | 切线与法线
The derivative at x = a is the gradient of the tangent to the curve y = f(x) at the point (a, f(a)). The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent gradient when the tangent gradient is not zero.
在x = a处的导数是曲线y = f(x)在点(a, f(a))处切线的斜率。法线与切线垂直，因此当切线斜率不为零时，法线斜率为切线斜率的负倒数。
tangent: y – f(a) = m(x – a), where m = f'(a)
normal gradient = -1/m, if m ≠ 0
In exam questions, first find f'(x), substitute x = a to get the gradient, then use the straight-line equation. If f'(a) = 0, the tangent is horizontal and the normal is vertical.
在考试题中，先求f'(x)，代入x = a得到斜率，再使用直线方程。如果f'(a) = 0，切线为水平线，法线为竖直线。
5. Stationary Points | 驻点
Stationary points occur where the derivative is zero. At these points the tangent to the curve is horizontal. Solving f'(x) = 0 gives the x-coordinates of the stationary points.
驻点出现在导数为零的位置。在这些点处，曲线的切线为水平线。求解f'(x) = 0可得驻点的x坐标。
dy/dx = 0
Stationary points can be classified as local maxima, local minima, or points of inflection. A local maximum has a decreasing gradient across the point, while a local minimum has an increasing gradient across the point.
驻点可分为局部极大值、局部极小值或拐点。局部极大值处越过该点梯度由正变负，局部极小值处梯度由负变正。
6. Second Derivative and Nature of Turning Points | 二阶导数与转折点性质
The second derivative evaluates the concavity of the curve and can be used to classify a stationary point. If f”(a) > 0, the point is a local minimum; if f”(a) < 0, the point is a local maximum.
二阶导数用于判断曲线的凹凸性，并可用于对驻点进行分类。如果f”(a) > 0，该点为局部极小值；如果f”(a) < 0，该点为局部极大值。
If f”(a) > 0 → local minimum; If f”(a) < 0 → local maximum
If f”(a) = 0, the second derivative test is inconclusive. You should then use a sign table for dy/dx on either side of the stationary point to decide its nature.
如果f”(a) = 0，二阶导数检验无法得出结论。此时应当在驻点两侧对dy/dx作符号表，以判断驻点性质。
| Condition | Nature | 中文 |
| f”(a) > 0 | Local minimum | 局部极小值 |
| f”(a) < 0 | Local maximum | 局部极大值 |
| f”(a) = 0 | Use sign test | 使用符号检验 |
7. Increasing and Decreasing Functions | 增函数与减函数
A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0. This relationship is often tested by asking you to find intervals on which a curve is increasing or decreasing.
函数在f'(x) > 0的区间上递增，在f'(x) < 0的区间上递减。这一关系常以求解曲线递增或递减区间的形式进行考查。
f'(x) > 0 ⇒ f is increasing
f'(x) < 0 ⇒ f is decreasing
You may also be given a derivative graph and asked to identify where the original function is increasing or decreasing. A value of f'(x) = 0 marks a boundary between these behaviours.
你还可能拿到导函数图像，要求判断原函数在哪里递增或递减。f'(x) = 0的点是这些行为之间的分界。
8. Optimisation Problems | 优化问题
Many Edexcel exam questions ask you to model a real-world quantity, express it in terms of one variable, differentiate, and set the derivative equal to zero to find the optimum value.
许多Edexcel考试题要求对现实量建模，将其表示为单个变量的函数，求导并令导数等于零，以求出最优值。
Steps: identify the quantity to optimise, write it as a function of one variable using given constraints, differentiate, solve f'(x) = 0, and justify the maximum or minimum using the second derivative or a sign table.
步骤：确定需要优化的量，利用给定约束将其写成一个变量的函数，求导，解f'(x) = 0，并用二阶导数或符号表说明是最大值还是最小值。
Always check the practical domain of the variable, such as x > 0, because endpoints may give maximum or minimum values even if they are not stationary points.
一定要检查变量的实际取值范围，例如x > 0，因为端点可能会给出最大值或最小值，即使它们不是驻点。
9. Rates of Change and Kinematics | 变化率与运动学
Differentiation is used to model rates of change. If displacement is given by s(t), then velocity is the first derivative and acceleration is the second derivative with respect to time.
微分用于对变化率建模。若位移为s(t)，则速度是位移关于时间的一阶导数，加速度是二阶导数。
v = ds/dt, a = dv/dt = d²s/dt²
In connected rates of change problems, the chain rule is used to link two rates. For example, if y is a function of x and x is a function of t, then dy/dt = dy/dx × dx/dt.
在相关变化率问题中，使用链式法则联系两个变化率。例如，如果y是x的函数且x是t的函数，则dy/dt = dy/dx × dx/dt。
Edexcel questions often ask you to find the maximum velocity or the time at which acceleration is zero, so you should be confident differentiating powers, roots, and simple rational functions.
Edexcel题目常要求求解最大速度或加速度为零的时刻，因此你应该熟练对幂函数、根式和简单有理函数求导。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
When differentiating, do not forget that a constant term differentiates to zero. Many students mistakenly keep the constant in the derivative.
求导时，不要忘记常数项求导为零。许多学生错误地在导数中保留了常数项。
Be careful when differentiating negative and fractional powers. For example, d/dx(1/x²) = -2x⁻³, not -2/x³ without simplifying to negative exponent form if the answer requires a single fraction.
求负指数和分数指数时要小心。例如，d/dx(1/x²) = -2x⁻³，如果答案要求写成一个分式，不要忘记化简。
When finding a normal, take the negative reciprocal of the tangent gradient, not the same gradient. Also, for points of inflection, show a sign change of dy/dx rather than relying on f”(x) = 0 alone.
求法线时，取切线斜率的负倒数，而不是相同斜率。此外，对于拐点，要证明dy/dx的符号发生改变，而不能仅仅依据f”(x) = 0。
Always show full working when solving f'(x) = 0, and classify the stationary point clearly. In modelling questions, check that your answer makes sense in context.
解f'(x) = 0时始终展示完整过程，并清楚地判断驻点类型。在建模题中，检查答案在实际情境中是否合理。
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