📚 Applications of Differentiation | 微分的应用
Differentiation is one of the central pillars of A-Level Mathematics. Beyond the mechanics of finding derivatives, its true power lies in modelling real-world situations and analysing the behaviour of functions. This article explores the key applications of differentiation required for the Edexcel A-Level Mathematics course, from tangents and normals to optimisation and kinematics.
微分是 A-Level 数学的核心支柱之一。除了掌握求导技巧，微分的真正威力在于对现实世界进行建模并分析函数的行为。本文探讨 Edexcel A-Level 数学课程中微分的核心应用，从切线与法线到最优化和运动学。
1. Tangents and Normals | 切线与法线
t point. If the point is (x₁, y₁), the tangent line can be found using y − y₁ = m(x − x₁).
曲线上某点的导数 dy/dx 给出了该点处切线的斜率。若该点为 (x₁, y₁)，则切线方程可由 y − y₁ = m(x − x₁) 求得。
The normal to a curve is perpendicular to the tangent. Its gradient is the negative reciprocal of the tangent's gradient, i.e. mnormal = −1 / m_tangent, provided m_tangent ≠ 0.
曲线的法线与切线垂直，其斜率为切线斜率的负倒数，即 m_法线 = −1 / m_切线，前提是 m_切线 ≠ 0。
- Find dy/dx and evaluate it at x = x₁ to get m_tangent.
- 计算 dy/dx 并在 x = x₁ 处求值得到 m_切线。
- = m_tangent (x − x₁) for the tangent; for the normal, replace gradient with −1/m_tangent.
- 利用 y − y₁ = m_切线 (x − x₁) 写出切线方程；法线方程则用梯度 −1/m_切线。
2. Stationary Points | 驻点
, local minima, or points of inflection. The first derivative test examines the sign of dy/dx on either side to classify the stationary point.
驻点出现在 dy/dx = 0 处。这些点可以是局部极大值、局部极小值或拐点。一阶导数检验通过考察两边 dy/dx 的符号来判断驻点类型。
If dy/dx 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 likely a point of inflection.
若 dy/dx 由正变负，则为局部极大值；若由负变正，则为局部极小值；若符号不变，则很可能是一个拐点。
3. Increasing and Decreasing Functions | 增函数与减函数
The sign of the derivative determines whether a function is increasing or decreasing on an interval. f'(x) > 0 implies the function is strictly increasing; f'(x) < 0 implies it is strictly decreasing.
导数的符号决定了函数在区间上是递增还是递减。f'(x) > 0 表示函数严格递增；f'(x) < 0 表示严格递减。
To find intervals of increase or decrease, solve f'(x) > 0 and f'(x) < 0. The endpoints of these intervals are typically stationary points or points where the derivative is undefined.
为确定递增或递减区间，解不等式 f'(x) > 0 与 f'(x) < 0。区间端点通常是驻点或导数不存在的点。
4. Second Derivative Test | 二阶导数检验
e second derivative provides an alternative method for classifying stationary points. If f'' > 0, the function is concave up at x = a and the stationary point is a local minimum. If f''(a) < 0, it is concave down and the point is a local maximum.
二阶导数提供了分类驻点的另一种方法。若 f''(a) > 0，函数在 x = a 处下凸，驻点为局部极小值；若 f''(a) < 0，则上凸，驻点为局部极大值。
When f'' nature of the stationary point must then be determined using the first derivative test or higher-order derivatives.
当 f''(a) = 0 时，二阶导数检验无法得出结论，必须使用一阶导数检验或更高阶导数来确定驻点性质。
5. Optimization Problems | 最优化问题
Many real-world problems involve maximising or minimising a quantity, such as area, volume, cost, or profit. Start by expressing the quantity to be optimised as a function of a single variable, often using a constraint equation to eliminate other variables.
许多实际问题涉及最大化或最小化某一量，如面积、体积、成本或利润。首先将待优化的量表示为单一变量的函数，通常借助约束方程消去其他变量。
Differentiate the function, set the derivative to zero to find critical points, and then use the second derivative test or the context to confirm whether the point gives a maximum or minimum. Always check the endpoints of the domain in closed-interval problems.
对函数求导，令导数为零以找到关键点，然后利用二阶导数检验或实际背景确认该点给出的是最大值还是最小值。对于闭区间问题，务必检查区间端点。
6. Rates of Change | 变化率
isplacement with respect to time, i.e. v = ds/dt, and acceleration is a = dv/dt = d²s/dt².
导数 dy/dx 可解释为 y 关于 x 的变化率。在物理情境中，速度是位移随时间的变化率，即 v = ds/dt；加速度 a = dv/dt = d²s/dt²。
radius is dA/dr = 2πr. This idea extends to any variables that are related by a function.
对于函数 A(r) = πr²，面积关于半径的变化率为 dA/dr = 2πr。这一思想可推广到任何由函数关联的变量上。
7. Connected Rates of Change | 关联变化率
depends on its side length x, and x is changing with time t, then dV/dt = (dV/dx)·(dx/dt).
当两个或多个变量随时间变化且存在方程关联时，链式法则给出了关联变化率。例如，若立方体体积 V 取决于边长 x，且 x 随时间 t 变化，则 dV/dt = (dV/dx)·(dx/dt)。
ease of surface area when the radius is increasing at a known rate. Set up dS/dt = (dS/dr)·(dr/dt) using S = 4πr².
确定已知变化率，表达要求的率，并运用链式法则。典型问题：一个球形气球被充气，已知半径变化率，求表面积增加率。利用 S = 4πr²，建立 dS/dt = (dS/dr)·(dr/dt)。
8. Sketching the Gradient Function | 绘制导函数图像
Given the graph of y = f(x), it is possible to sketch the gradient function y = f'x) without knowing the algebraic form of f. At points where f(x) has a stationary point, f'(x) = 0, so the gradient graph crosses the x-axis.
给定 y = f(x) 的图像，即使不知道 f 的代数形式，也可以绘制其导函数 y = f'(x) 的草图。在 f(x) 的驻点处，f'(x) = 0，所以导函数图像穿过 x 轴。
Where f(x) is increasing, f'(x) is above the x-axis; where f(x) is decreasing, f'(x) is below the x-axis. The steepness of the original graph corresponds to the height of the gradient graph.
当 f(x) 递增时，f'(x) 在 x 轴上方；当 f(x) 递减时，f'(x) 在 x 轴下方。原图的陡峭程度对应导函数图像的高度。
9. Further Applications – Kinematics | 进一步应用 – 运动学
), velocity v(t) and acceleration a(t) are linked by differentiation. Given s(t), v(t) = s'(t) and a(t) = v'(t) = s''(t). This is vital for solving problems involving motion along a line.
在运动学中，位移 s(t)、速度 v(t) 和加速度 a(t) 通过微分关联。给定 s(t)，有 v(t) = s'(t) 以及 a(t) = v'(t) = s''(t)。这对于解决直线运动问题至关重要。
irection, set v(t) = 0. To find maximum speed, differentiate the speed function or analyse |v(t)| indicates whether the particle is accelerating or decelerating relative to its direction.
求最大位移或质点何时改变方向，令 v(t) = 0。求最大速率，可对速率函数求导或分析 |v(t)|。a(t) 的符号表示质点相对于其运动方向是在加速还是减速。
which is useful when acceleration is given as a function of displacement rather than time.
利用链式法则，加速度也可写为 a = v dv/ds，当加速度被表示为位移的函数而非时间函数时，这种形式非常有用。
10. Exam Tips and Common Pitfalls | 考试技巧与常见错误
Always simplify the function as much as possible before differentiating, especially when dealing with rational or root forms. For example, rewrite 1/x² as x⁻² and √x as x1/2 before applying standard rules.
在求导前尽可能化简函数，特别是处理有理式或根式时。例如，先将 1/x² 写成 x⁻²，√x 写成 x1/2，再应用标准求导法则。
Be careful with the chain, product and quotient rules. Label intermediate steps clearly to avoid errors. For connected rates, explicitly write out the chain rule with the correct derivatives before substituting values.
谨慎使用链式法则、乘积和商法则。清晰标注中间步骤以避免错误。处理关联变化率时，先明确写出含有正确导数的链式法则，再代入数值。
Double-check the classification of stationary points using both first and second derivative tests where possible. Always consider the domain of the function, especially endpoints, in optimisation problems.
尽可能用一阶和二阶导数检验双重确认驻点分类。在最优化问题中，务必考虑函数的定义域，尤其是端点。
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