📚 Rates of Change | 变化率
Rates of change lie at the heart of differentiation. In A-Level Mathematics, you are expected to move from finding gradients of curves to modelling real-world situations where one quantity changes with respect to another, often time. This article covers average and instantaneous rates, connected rates, parametric and implicit differentiation, exponential models, and kinematics, with Edexcel-style examples.
变化率是微分的核心。在 A-Level 数学中，你需要从求曲线的梯度逐步过渡到对现实情境建模，即一个量相对于另一个量（通常是时间）发生变化。本文涵盖平均变化率与瞬时变化率、关联变化率、参数方程与隐函数求导、指数模型以及运动学，并配有 Edexcel 风格的例题。
1. Average and Instantaneous Rate of Change | 平均变化率与瞬时变化率
te of change over the interval [a, b] is the gradient of the chord joining (a, f(a)) and (b, f(b)). It measures the overall change in y per unit change in x across that interval.
对于函数 y = f(x)，在区间 [a, b] 上的平均变化率是连接 (a, f(a)) 和 (b, f(b)) 的割线斜率。它表示在该区间内每单位 x 变化对应的 y 的总变化。
Average rate of change = Δy / Δx = [f(b) − f(a)] / (b − a)
he derivative dy/dx at x = a, and it gives the gradient of the tangent line at that point.
在 x = a 处的瞬时变化率通过让 b 趋近于 a 来求取。这个极限就是 x = a 处的导数 dy/dx，它给出该点处切线的斜率。
A speedometer reading is a familiar practical example: it shows the instantaneous rate of change of distance with respect to time. In contrast, average speed over a journey uses total distance divided by total time.
速度表的读数就是一个熟悉的实际例子：它表示距离相对于时间的瞬时变化率。相比之下，一段旅程的平均速度则用总距离除以总时间。
Instantaneous rate = dy / dx = lim (Δx → 0) Δy / Δx
2. The Derivative as a Gradient Function | 导数作为梯度函数
For y = f(x), the derivative f' oint x. Differentiation from first principles uses the limit definition of the derivative.
对于 y = f(x)，导数 f'(x) 或 dy/dx 给出曲线在任意点 x 处的梯度。从基本原理出发的微分使用导数的极限定义。
f'(x) = lim (h → 0) [f(x + h) − f(x)] / h
negative and fractional powers, provided the function is differentiable at the point considered.
幂法则是 A-Level 中最常用的法则：若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。只要函数在考虑的点可导，该法则对正幂、负幂和分数幂都适用。
− 4 = 8. This tells you the tangent at x = 2 rises 8 units for every 1 unit increase in x.
例如，若 y = 3x² − 4x + 1，则 dy/dx = 6x − 4。当 x = 2 时，梯度为 6(2) − 4 = 8。这说明 x = 2 处的切线每增加 1 个单位的 x，就上升 8 个单位。
3. Notation and the Chain Rule Preview | 记法与链式法则预告
Leibniz notation dy/dx is especially useful for rates and the chain rule because it shows the dependent and independent variables explicitly. Function notation f'(x) is more compact and is often used when evaluating gradients at specific points.
莱布尼茨记法 dy/dx 对变化率和链式法则特别有用，因为它明确显示出自变量和因变量。函数记法 f'(x) 更简洁，常用于在特定点求梯度值。
) × (du/dx). This rule is essential for composite functions and connected rates of change.
链式法则指出，若 y 是 u 的函数，而 u 是 x 的函数，则 dy/dx = (dy/du) × (du/dx)。该法则对复合函数和关联变化率至关重要。
, let u = 2x + 1. Then dy/du = 5u⁴ and du/dx = 2, so dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴.
例如，对于 y = (2x + 1)⁵，令 u = 2x + 1。则 dy/du = 5u⁴，du/dx = 2，所以 dy/dx = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴。
4. Increasing, Decreasing and Stationary Points | 增减性与驻点
A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0. Stationary points occur where f'(x) = 0, meaning the tangent is horizontal.
函数在 f'(x) > 0 的区间上递增，在 f'(x) < 0 的区间上递减。驻点出现在 f'(x) = 0 处，即切线水平。
To classify a stationary point, you can examine the sign of f'(x) on either side, or use the second derivative test. A change from positive to negative gradient indicates a local maximum; negative to positive indicates a local minimum.
要判断驻点的类型，可以检查 f'(x) 在两侧的符号，或使用二阶导数检验。梯度由正变负表示局部极大值；由负变正表示局部极小值。
= −1, it is −6, so this is a local maximum. At x = 1, it is 6, so this is a local minimum.
例如，对于 y = x³ − 3x，dy/dx = 3x² − 3 = 3(x − 1)(x + 1)。驻点出现在 x = −1 和 x = 1。二阶导数为 d²y/dx² = 6x。在 x = −1 处为 −6，所以是局部极大值；在 x = 1 处为 6，所以是局部极小值。
5. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative d²y/dx² is the rate of change of the gradient. It tells you how the slope itself is changing as x increases.
二阶导数 d²y/dx² 是梯度的变化率。它告诉你斜率本身随 x 增大如何变化。
ocal maximum. If d²y/dx² = 0, the test is inconclusive and you must examine the sign of the first derivative.
若在驻点处 d²y/dx² > 0，则曲线开口向上，该点为局部极小值。若 d²y/dx² < 0，则曲线开口向下，该点为局部极大值。若 d²y/dx² = 0，则检验无结论，必须检查一阶导数的符号。
Concavity also helps in curve sketching and in understanding the behaviour of rates: a positive second derivative of displacement means acceleration is positive, for instance.
凹凸性也有助于曲线绘制和理解变化率的行为：例如，位移的二阶导数为正表示加速度为正。
6. Connected Rates of Change | 关联变化率
ies that change with time. The chain rule connects their rates: dy/dt = (dy/dx) × (dx/dt).
关联变化率问题涉及两个或多个随时间变化的量。链式法则连接它们的变化率：dy/dt = (dy/dx) × (dx/dt)。
A reliable method is: write down the rate you know, write down the rate you want, find an equation linking the variables, differentiate with respect to time, then substitute the known values.
一个可靠的方法是：写下已知变化率，写下要求的变化率，找到连接变量的方程，对时间求导，然后代入已知值。
Since V = (4/3)πr³, dV/dr = 4πr². Then dV/dt = dV/dr × dr/dt = 4π(10)² × 0.5 = 200π cm³/s.
例：一个气球被充气，半径以 0.5 cm/s 增加。求 r = 10 cm 时体积的增加率。因为 V = (4/3)πr³，所以 dV/dr = 4πr²。于是 dV/dt = dV/dr × dr/dt = 4π(10)² × 0.5 = 200π cm³/s。
dV / dt = (dV / dr) × (dr / dt)
7. Implicit Differentiation and Rates | 隐函数求导与变化率
uch cases, differentiate both sides with respect to x, remembering that y is a function of x.
有些 x 与 y 之间的关系不是显式地写成 y = f(x)。在这种情况下，对两边关于 x 求导，并记住 y 是 x 的函数。
/dx = −x/y. This gives the gradient of the tangent to the circle at any point where y ≠ 0.
例如，对于 x² + y² = 25，逐项求导得 2x + 2y(dy/dx) = 0。整理得 dy/dx = −x/y。这给出圆在任意 y ≠ 0 点处的切线梯度。
Implicit differentiation is also useful when variables are related by an equation that is difficult or impossible to rearrange. In connected rates, it allows you to differentiate a relation directly with respect to time.
当变量之间的关系方程难以或无法整理成显函数时，隐函数求导也很有用。在关联变化率中，它允许你直接对关系式关于时间求导。
8. Parametric Differentiation | 参数方程求导
erms of a parameter t, the chain rule gives dy/dx = (dy/dt) ÷ (dx/dt), provided dx/dt ≠ 0.
当 x 和 y 用参数 t 表示时，链式法则给出 dy/dx = (dy/dt) ÷ (dx/dt)，前提是 dx/dt ≠ 0。
− 1, so dy/dx = (3t² − 1) / (2t). This expresses the gradient in terms of the parameter t.
例：x = t²，y = t³ − t。则 dx/dt = 2t，dy/dt = 3t² − 1，所以 dy/dx = (3t² − 1) / (2t)。这用参数 t 表示了梯度。
To find the second derivative d²y/dx², differentiate dy/dx with respect to t, then divide by dx/dt. This is a common extension question in A-Level exams.
要求二阶导数 d²y/dx²，先对 t 求 dy/dx 的导数，再除以 dx/dt。这是 A-Level 考试中常见的拓展题。
9. Exponential Models and Rates of Change | 指数模型与变化率
Exponential models have the form y = A eᵏˣ constant. A key property is that the rate of change is proportional to the current value.
指数模型的形式为 y = A eᵏˣ，其中 A 是初始值，k 是速率常数。一个关键性质是变化率与当前值成正比。
If y = A eᵏˣ, then dy / dx = kA eᵏˣ = k y
If k > 0, the quantity grows exponentially; if k < 0, it decays exponentially. This model appears in population growth, radioactive decay, and Newton's law of cooling.
若 k > 0，该量呈指数增长；若 k < 0，则呈指数衰减。该模型出现在人口增长、放射性衰变和牛顿冷却定律中。
For example, if y = 200 e0.05t, then dy/dt = 10 e0.05t = 0.05y. At t = 0, the rate of change is 10 units per unit time, which is 5% of the initial value.
例如，若 y = 200 e0.05t，则 dy/dt = 10 e0.05t = 0.05y。在 t = 0 时，变化率为每单位时间 10 个单位，即初始值的 5%。
10. Kinematics: Displacement, Velocity, Acceleration | 运动学：位移、速度、加速度
displacement is s(t), then velocity is v = ds/dt and acceleration is a = dv/dt = d²s/dt².
在运动学中，变化率连接位移、速度和加速度。若位移为 s(t)，则速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。
Speed is the magnitude of velocity, |v| 0. Maximum or minimum velocity occurs when a = 0, provided the acceleration changes sign.
速率是速度的大小 |v|。当 v = 0 时，物体瞬时静止。当 a = 0 且加速度变号时，速度达到最大或最小值。
elocity is at a local minimum or maximum. Substituting t = 2 gives v = 3(4) − 24 + 9 = −3.
例：s = t³ − 6t² + 9t。则 v = 3t² − 12t + 9 = 3(t − 1)(t − 3)，a = 6t − 12。物体在 t = 1 和 t = 3 时静止。在 t = 2 时，a = 0，所以速度达到局部最小或最大值。代入 t = 2 得 v = 3(4) − 24 + 9 = −3。
11. Exam-Style Worked Examples | 考试风格例题
The y-coordinate is 8 − 12 + 2 = −2. Therefore the tangent is the horizontal line y = −2.
例 1：求 y = x³ − 3x² + 2 在 x = 2 处的切线方程。求导得 dy/dx = 3x² − 6x。在 x = 2 处，梯度为 3(4) − 12 = 0。y 坐标为 8 − 12 + 2 = −2。因此切线是水平线 y = −2。
ncrease of area when r = 5 cm. A = πr², so dA/dr = 2πr. Then dA/dt = 2π(5)(2) = 20π cm²/s.
例 2：圆的半径以 2 cm/s 增加。求 r = 5 cm 时面积的增加率。A = πr²，所以 dA/dr = 2πr。于是 dA/dt = 2π(5)(2) = 20π cm²/s。
Example 3: A curve has parametric equations x = t², y = t³ − t. Find
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