Differentiation | 微分

📚 Differentiation | 微分

Differentiation is one of the core topics in the Edexcel A-Level Mathematics specification. It allows us to measure how a function changes at any point, giving the gradient of a tangent and enabling analysis of rates of change, optimisation and curve sketching.

微分是 Edexcel A-Level 数学考纲中的核心主题之一。它使我们能够度量函数在任意点的变化情况，给出切线的斜率，并用于变化率、最优化和曲线图像分析。

1. First Principles and the Derivative | 第一原理与导数

The derivative of a function f(x) at a point x is defined formally by the limit of the average rate of change as the interval h tends to zero.

函数 f(x) 在点 x 处的导数由平均变化率在区间 h 趋于零时的极限正式定义。

f'(x) = limₕ→₀ [f(x + h) − f(x)] / h

This limit gives the gradient of the tangent to the curve y = f(x) at that point.

这个极限给出了曲线 y = f(x) 在该点处切线的斜率。

The notation f'(x), dy/dx and d/dx [f(x)] all represent the same derivative.

符号 f'(x)、dy/dx 和 d/dx [f(x)] 都表示同一个导数。

2. Power Rule and Basic Rules | 幂法则与基本法则

For any real power n, if y = xⁿ then dy/dx = n xⁿ⁻¹. This is the most frequently used rule in differentiation.

对于任意实数幂 n，若 y = xⁿ，则 dy/dx = n xⁿ⁻¹。这是微分中最常用的法则。

Differentiation is linear, so constants can be taken outside and sums can be differentiated term by term.

微分具有线性性质，因此常数可以提到外面，和式可以逐项求导。

d/dx [a f(x) + b g(x)] = a f'(x) + b g'(x)

For example, if y = 3x⁴ − 5x² + 2x − 7, then dy/dx = 12x³ − 10x + 2.

例如，若 y = 3x⁴ − 5x² + 2x − 7，则 dy/dx = 12x³ − 10x + 2。

3. Standard Derivatives | 常见函数的导数

The following standard derivatives must be memorised for Edexcel A-Level Mathematics.

以下常见函数的导数必须熟记，适用于 Edexcel A-Level 数学考试。

Function \| 函数	Derivative \| 导数
y = xⁿ	dy/dx = n xⁿ⁻¹
y = eˣ	dy/dx = eˣ
y = ln x	dy/dx = 1/x
y = sin x	dy/dx = cos x
y = cos x	dy/dx = −sin x
y = tan x	dy/dx = sec² x

Note the negative sign in the derivative of cos x: it is a common source of sign errors.

注意 cos x 导数中的负号：这是常见的符号错误来源。

4. Chain Rule | 链式法则

The chain rule is used when differentiating a composite function y = f(g(x)).

链式法则用于对复合函数 y = f(g(x)) 求导。

dy/dx = dy/du × du/dx

Here u = g(x) is the inner function. In words: differentiate the outer function, then multiply by the derivative of the inner function.

这里 u = g(x) 是内层函数。用文字表述：先对外层函数求导，再乘以内层函数的导数。

For example, if y = (3x² + 1)⁵, let u = 3x² + 1, then y = u⁵, dy/du = 5u⁴ and du/dx = 6x, so dy/dx = 30x(3x² + 1)⁴.

例如，若 y = (3x² + 1)⁵，令 u = 3x² + 1，则 y = u⁵，dy/du = 5u⁴ 且 du/dx = 6x，因此 dy/dx = 30x(3x² + 1)⁴。

5. Product Rule | 乘积法则

When y = u(x) v(x) is a product of two functions, the derivative is given by the product rule.

当 y = u(x) v(x) 是两个函数的乘积时，其导数由乘积法则给出。

dy/dx = u dv/dx + v du/dx

A useful memory aid is “first times derivative of second plus second times derivative of first”.

一个有用的记忆方法是：「前乘后的导数加后乘前的导数」。

For example, if y = x² sin x, then dy/dx = x² cos x + 2x sin x.

例如，若 y = x² sin x，则 dy/dx = x² cos x + 2x sin x。

6. Quotient Rule | 商法则

When y = u(x) / v(x) is a quotient of two functions, use the quotient rule.

当 y = u(x) / v(x) 是两个函数的商时，使用商法则。

dy/dx = (v du/dx − u dv/dx) / v²

The numerator must start with v du/dx, not u dv/dx; reversing the order changes the sign.

分子必须从 v du/dx 开始，而不是 u dv/dx；顺序颠倒会改变符号。

For example, if y = x / (x² + 1), then dy/dx = [(x² + 1)(1) − x(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)².

例如，若 y = x / (x² + 1)，则 dy/dx = [(x² + 1)(1) − x(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)²。

7. Second Derivatives | 二阶导数

The second derivative is obtained by differentiating dy/dx again. It is written f”(x) or d²y/dx².

二阶导数是对 dy/dx 再次求导得到的。写作 f”(x) 或 d²y/dx²。

The second derivative measures the rate of change of the gradient and helps determine concavity and the nature of stationary points.

二阶导数度量斜率的变化率，并帮助判断凹凸性和驻点的性质。

If d²y/dx² > 0, the curve is concave up (a minimum may occur); if d²y/dx² < 0, the curve is concave down (a maximum may occur).

若 d²y/dx² > 0，曲线向上凹（可能出现最小值）；若 d²y/dx² < 0，曲线向下凹（可能出现最大值）。

For example, if y = x³ − 3x² + 2, then dy/dx = 3x² − 6x and d²y/dx² = 6x − 6.

例如，若 y = x³ − 3x² + 2，则 dy/dx = 3x² − 6x，d²y/dx² = 6x − 6。

8. Stationary Points and Turning Points | 驻点与拐点

A stationary point occurs where dy/dx = 0. At a stationary point, the tangent to the curve is horizontal.

驻点出现在 dy/dx = 0 处。在驻点处，曲线的切线是水平的。

To classify a stationary point, substitute its x-coordinate into the second derivative: d²y/dx² > 0 gives a local minimum, d²y/dx² < 0 gives a local maximum.

要判断驻点类型，将其 x 坐标代入二阶导数：d²y/dx² > 0 得到局部最小值，d²y/dx² < 0 得到局部最大值。

If d²y/dx² = 0, the test is inconclusive and you should examine the sign of dy/dx on either side of the point.

若 d²y/dx² = 0，该判别法无法确定，应检查该点两侧 dy/dx 的符号。

For example, for y = x³ − 3x, dy/dx = 3x² − 3. Setting dy/dx = 0 gives x = ±1. d²y/dx² = 6x, so x = −1 is a maximum and x = 1 is a minimum.

例如，对于 y = x³ − 3x，dy/dx = 3x² − 3。令 dy/dx = 0 得 x = ±1。d²y/dx² = 6x，因此 x = −1 处为最大值，x = 1 处为最小值。

9. Tangents and Normals | 切线与法线

The gradient of the tangent at a point x = a is m = f'(a). The equation of the tangent is y − y₁ = m(x − x₁).

在 x = a 处切线的斜率为 m = f'(a)。切线方程为 y − y₁ = m(x − x₁)。

The normal is perpendicular to the tangent, so its gradient is −1/m, provided m ≠ 0.

法线垂直于切线，因此其斜率为 −1/m，前提是 m ≠ 0。

For example, if y = x², then dy/dx = 2x. At x = 1, m = 2 and y₁ = 1. The tangent is y − 1 = 2(x − 1), and the normal has gradient −1/2.

例如，若 y = x²，则 dy/dx = 2x。在 x = 1 处，m = 2 且 y₁ = 1。切线为 y − 1 = 2(x − 1)，法线斜率为 −1/2。

10. Optimisation | 最优化

Differentiation is widely used to find maximum or minimum values in real-world problems such as minimising cost, maximising area or maximising profit.

微分广泛用于求解现实世界中的最大值或最小值问题，例如最小化成本、最大化面积或最大化利润。

The method is: express the quantity to be optimised as a function of one variable, differentiate, set dy/dx = 0 to find stationary points, and classify using the second derivative or sign test.

方法为：将需要优化的量表示为单变量函数，求导，令 dy/dx = 0 求驻点，再用二阶导数或符号检验判断类型。

Always check the domain of the variable, as endpoints may also give optimal values.

务必检查变量的定义域，因为端点也可能取得最优值。

11. Rates of Change | 变化率

If a quantity y changes with time t, then dy/dt is its rate of change. Differentiation links related rates through the chain rule.

若量 y 随时间 t 变化，则 dy/dt 是其变化率。微分通过链式法则将相关变化率联系起来。

dy/dt = dy/dx × dx/dt

For example, if a spherical balloon expands and its radius r increases at 2 cm/s, the rate of change of volume is dV/dt = dV/dr × dr/dt = 4πr² × 2.

例如，若一个球形气球膨胀，其半径 r 以 2 cm/s 增加，则体积的变化率为 dV/dt = dV/dr × dr/dt = 4πr² × 2。

12. Parametric and Implicit Differentiation | 参数方程与隐函数微分

For parametric equations x = f(t), y = g(t), the derivative is dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

对于参数方程 x = f(t)、y = g(t)，其导数为 dy/dx = (dy/dt) / (dx/dt)，前提是 dx/dt ≠ 0。

For implicit equations such as x² + y² = 25, differentiate both sides with respect to x, treating y as a function of x.

对于隐式方程，如 x² + y² = 25，对两边关于 x 求导，将 y 视为 x 的函数。

For x² + y² = 25, differentiating gives 2x + 2y dy/dx = 0, so dy/dx = −x/y.

对于 x² + y² = 25，求导得 2x + 2y dy/dx = 0，因此 dy/dx = −x/y。

13. Common Mistakes and Exam Tips | 常见错误与考试技巧

Do not forget to multiply by the derivative of the inner function when using the chain rule.

使用链式法则时，不要忘记乘以内层函数的导数。

In the quotient rule, the numerator must be v du/dx − u dv/dx, not the reverse.

在商法则中，分子必须是 v du/dx − u dv/dx，而不是颠倒顺序。

Always simplify your derivative before finding stationary points, as this reduces algebraic errors.

在求驻点之前，务必先化简导数，这样可以减少代数错误。

When an optimisation question gives a fixed perimeter or area, use the constraint to eliminate one variable before differentiating.

当最优化问题给出固定的周长或面积时，利用约束条件消去一个变量，然后再求导。

Show clear working and use correct notation throughout, as Edexcel examiners award method marks even if the final answer is wrong.

全程展示清晰步骤并使用正确符号，因为即使最终答案错误，Edexcel 考官仍会给予方法分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Differentiation | 微分