📚 Using Differentiation | 导数的应用
In Edexcel A Level Mathematics, differentiation is more than a mechanical skill. It gives the gradient of a curve at any point, describes rates of change, and provides tools for optimisation, motion, and modelling. This guide shows how to use differentiation effectively across typical exam questions.
在 Edexcel A Level 数学中，微分法不仅仅是一种机械运算。它给出了曲线在任意点的梯度，描述变化率，并为优化、运动学和建模提供工具。本指南展示如何在典型考试题目中有效运用导数。
1. The Derivative as a Rate of Change | 导数作为变化率
For a function y = f(x), the derivative dy/dx or f'(x) gives the instantaneous rate of change of y with respect to x. It is the gradient of the tangent to the curve y = f(x) at a given point.
对于函数 y = f(x)，导数 dy/dx 或 f'(x) 表示 y 关于 x 的瞬时变化率。它是曲线 y = f(x) 在给定点处切线的梯度。
If dy/dx > 0, y is increasing as x increases; if dy/dx < 0, y is decreasing. This sign interpretation underpins most applications of differentiation.
如果 dy/dx > 0，则 y 随 x 增大而递增；如果 dy/dx < 0，则 y 随 x 增大而递减。这种符号解释是导数大多数应用的基础。
dy/dx = lim (Δx → 0) [f(x + Δx) – f(x)] / Δx
2. Finding Tangents and Normals | 求切线与法线
To find the gradient at a point x = a, first differentiate f(x), then substitute x = a into f'(x). This value m is the gradient of the tangent at that point.
为了求 x = a 处的梯度，先对 f(x) 求导，再将 x = a 代入 f'(x)。所得值 m 就是该点处切线的梯度。
The normal is perpendicular to the tangent. Its gradient is -1/m, provided m ≠ 0. If the tangent is horizontal, the normal is vertical.
法线垂直于切线。其梯度为 -1/m（当 m ≠ 0 时）。如果切线是水平的，法线就是竖直的。
With a point (x₁, y₁) and gradient m, the tangent equation is y – y₁ = m(x – x₁). The normal equation uses gradient -1/m.
已知点 (x₁, y₁) 和梯度 m，切线方程为 y – y₁ = m(x – x₁)。法线方程使用梯度 -1/m。
3. Increasing and Decreasing Functions | 函数的递增与递减
A differentiable function is increasing on an interval where f'(x) ≥ 0, and decreasing where f'(x) ≤ 0. In exam solutions, state the intervals using x values and strict inequalities where appropriate.
可导函数在 f'(x) ≥ 0 的区间上递增，在 f'(x) ≤ 0 的区间上递减。在考试解答中，要用 x 值表示区间，并在适当处使用严格不等式。
To determine these intervals, solve f'(x) > 0 or f'(x) < 0. A sign diagram for f'(x) is often the clearest way to present the result.
要确定这些区间，解不等式 f'(x) > 0 或 f'(x) < 0。通常用 f'(x) 的符号图来清晰展示结果。
4. Stationary Points | 驻点
Stationary points occur where f'(x) = 0. At such points the tangent to the curve is horizontal. There are three possible natures: local maximum, local minimum, or point of inflection.
驻点出现在 f'(x) = 0 处。在这些点，曲线的切线是水平的。驻点有三种可能的性质：局部极大值、局部极小值或拐点。
Finding stationary points involves solving f'(x) = 0 for x, then substituting back into y = f(x) to obtain the y-coordinate.
求驻点需要解方程 f'(x) = 0 得到 x，然后代回 y = f(x) 求出 y 坐标。
5. Classifying Stationary Points | 驻点的分类
One way to classify a stationary point is to examine the sign of f'(x) on either side of the point. If f'(x) changes from positive to negative, the point is a local maximum. If it changes from negative to positive, it is a local minimum. If the sign does not change, it is a point of inflection.
对驻点进行分类的一种方法是考察该点两侧 f'(x) 的符号。如果 f'(x) 由正变负，该点是局部极大值；如果由负变正，则是局部极小值；如果符号不变，则是拐点。
Alternatively, the second derivative test uses f”(x). If f”(x) < 0, the point is a maximum; if f”(x) > 0, it is a minimum. If f”(x) = 0, the test is inconclusive and the sign of f'(x) must be used.
另一种方法是使用二阶导数判别法：若 f”(x) < 0，则为极大值；若 f”(x) > 0，则为极小值；若 f”(x) = 0，则判别法失效，必须使用 f'(x) 的符号进行判断。
Example: f(x) = x³ – 3x. Then f'(x) = 3x² – 3 = 3(x² – 1). Setting f'(x) = 0 gives x = ±1. f”(x) = 6x, so f”(1) = 6 > 0, indicating a minimum at x = 1, and f”(-1) = -6 < 0, indicating a maximum at x = -1.
例题：f(x) = x³ – 3x。则 f'(x) = 3x² – 3 = 3(x² – 1)。令 f'(x) = 0 得 x = ±1。f”(x) = 6x，所以 f”(1) = 6 > 0，说明 x = 1 处为极小值；f”(-1) = -6 < 0，说明 x = -1 处为极大值。
| f'(x) changes from + to – | f'(x) 由正变负：局部极大值 |
| f'(x) changes from – to + | f'(x) 由负变正：局部极小值 |
| f'(x) does not change sign | f'(x) 符号不变：拐点 |
6. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative f”(x) describes the rate of change of the gradient, and therefore the curvature of the graph. Where f”(x) > 0, the curve is convex (or concave up); where f”(x) < 0, it is concave (or concave down).
二阶导数 f”(x) 描述梯度的变化率，因此反映图像的弯曲方向。当 f”(x) > 0 时，曲线是下凸（凹向上）；当 f”(x) < 0 时，曲线是上凸（凹向下）。
Points where concavity changes are inflection points. Not all inflection points are stationary; some occur where f'(x) is non-zero but f”(x) changes sign.
凹凸性改变的点是拐点。并非所有拐点都是驻点；有些拐点出现在 f'(x) 不为零但 f”(x) 变号的位置。
7. Optimisation Problems | 优化问题
Many maximisation and minimisation problems ask for the greatest or least value of a quantity such as area, volume, profit, or cost. The method is to express the quantity as a function of one variable, differentiate, set f'(x) = 0, and then justify the nature.
许多最大值和最小值问题要求求面积、体积、利润或成本等量的最大值或最小值。方法是将该量表示为单一变量的函数，求导，令 f'(x) = 0，然后证明其性质。
Always check the domain of the variable, because the maximum or minimum may occur at an endpoint rather than at a stationary point. Use the second derivative test or a sign test to confirm the nature.
始终检查变量的定义域，因为最大值或最小值可能出现在端点而非驻点。用二阶导数判别法或符号判别法确认其性质。
For example, if a closed box has volume V = x²h and a surface area constraint is given, express h in terms of x before differentiating.
例如，若封闭盒子的体积为 V = x²h，且给定表面积约束，则需先将 h 表示为 x 的函数再求导。
8. Connected Rates of Change | 相关变化率
In connected rates problems, one rate such as dy/dt is known, and another related rate dx/dt or dV/dt must be found. The chain rule links them: dy/dt = (dy/dx)(dx/dt).
在相关变化率问题中，已知一个变化率如 dy/dt，需要求另一个相关变化率如 dx/dt 或 dV/dt。链式法则将它们联系起来：dy/dt = (dy/dx)(dx/dt)。
Write down the known rates, identify the equation connecting the variables, differentiate both sides with respect to time t, then substitute the given values.
写出已知变化率，找出联系各变量的方程，对方程两边关于时间 t 求导，然后代入已知值。
dy/dt = (dy/dx) × (dx/dt)
9. Kinematics Applications | 运动学应用
For a particle moving in a straight line, displacement s(t), velocity v(t), and acceleration a(t) are related by differentiation: v = ds/dt, a = dv/dt = d²s/dt².
对于沿直线运动的质点，位移 s(t)、速度 v(t) 和加速度 a(t) 通过微分相联系：v = ds/dt，a = dv/dt = d²s/dt²。
A particle is at rest when v = 0. It changes direction when the sign of v changes, often at a time where v = 0 but the sign changes on either side.
当 v = 0 时质点静止。当 v 的符号改变时质点改变运动方向，通常出现在 v = 0 且两侧符号不同的时刻。
Maximum displacement from the origin occurs when v = 0 and the sign of v changes from positive to negative, or at an endpoint of the time interval.
质点相对原点的最大位移出现在 v = 0 且 v 由正变负处，或在时间区间的端点处。
10. Common Mistakes and Exam Tips | 常见错误与考试技巧
Common errors include incorrectly differentiating powers with fractional or negative indices, misapplying the chain rule, and forgetting to substitute back into the original function to find the y-coordinate of a stationary point.
常见错误包括：错误处理分数或负指数的幂函数求导、误用链式法则，以及忘记将 x 代回原函数来求驻点的 y 坐标。
When finding tangents, some students give only the gradient and omit the full equation. Always write the complete tangent or normal equation in the required form.
求切线时，有些学生只给出梯度而遗漏完整方程。必须始终按题目要求写出完整的切线或法线方程。
Show the derivative, the substituted gradient, the nature test, and the final answer with units where relevant. Present intervals and coordinates clearly.
展示导数、代入后的梯度、性质判别以及最终答案（必要时带单位）。清晰写出区间和坐标。
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