Differentiation: Rules, Techniques and Exam Applications | 微分：法则、技巧与考试应用

📚 Differentiation: Rules, Techniques and Exam Applications | 微分：法则、技巧与考试应用

Differentiation is one of the most important topics in Edexcel A-Level Mathematics. It underpins graph sketching, optimisation, kinematics and many modelling questions. This article covers the key rules, techniques and common exam applications you need to master.

微分是 Edexcel A-Level 数学中最重要的主题之一。它是函数作图、最优化、运动学以及许多建模问题的基础。本文涵盖你需要掌握的核心法则、技巧与常见考试应用。

1. First Principles and the Gradient Function | 第一原理与梯度函数

The derivative of a function f(x) is defined as the limit of the average rate of change as the interval tends to zero. This is called differentiation from first principles and is required by Edexcel for simple polynomials.

函数 f(x) 的导数定义为区间趋近于零时平均变化率的极限。这称为从第一原理求导，Edexcel 考试要求能对简单多项式使用这一方法。

f'(x) = lim(h→0) [f(x + h) − f(x)] / h

For example, if f(x) = x², expand (x + h)² = x² + 2xh + h². Subtracting f(x) and dividing by h gives 2x + h, and as h→0 the limit is 2x.

例如，若 f(x) = x²，展开 (x + h)² = x² + 2xh + h²。减去 f(x) 再除以 h 得到 2x + h，当 h→0 时极限为 2x。

Examiners often ask you to prove from first principles that the derivative of xⁿ is nxⁿ⁻¹ for small n. You must show full algebraic steps, not just state the result.

考官常要求你从第一原理证明 xⁿ 的导数是 nxⁿ⁻¹（n 较小时）。你必须展示完整的代数步骤，而不能只写出结果。

2. Power Rule and Basic Derivatives | 幂法则与基本导数

The power rule is the most frequently used differentiation tool. For any real constant n, the derivative of xⁿ is nxⁿ⁻¹. Constant multiples and sums can be differentiated term by term.

幂法则是最常用的求导工具。对于任意实数常量 n，xⁿ 的导数为 nxⁿ⁻¹。常数倍与和可以逐项求导。

d/dx (xⁿ) = nxⁿ⁻¹

The table below summarises the basic derivatives you must memorise for Edexcel Pure Mathematics. These results are used both directly and inside the chain rule.

下表总结了你在 Edexcel 纯数学中必须记住的基本导数。这些结果既可直接使用，也可用于链式法则内部。

Function	Derivative
xⁿ	nxⁿ⁻¹
sin x	cos x
cos x	−sin x
tan x	sec² x
eˣ	eˣ
ln x	1/x

You should also be comfortable with rewriting expressions before differentiating, such as 1/x = x⁻¹ and √x = x^½, so that the power rule can be applied.

你还应熟练在求导前改写表达式，例如 1/x = x⁻¹、√x = x^½，从而能够应用幂法则。

3. Rules: Product, Quotient and Chain Rules | 乘法、除法与链式法则

When differentiating products and quotients, you cannot simply differentiate each part separately. You must use the product rule or quotient rule. When differentiating a function of a function, you must use the chain rule.

对乘积和商求导时，不能简单地对每一部分分别求导。你必须使用乘法法则或除法法则。对复合函数求导时，必须使用链式法则。

If y = uv, where u and v are functions of x, then:

若 y = uv，其中 u 和 v 都是 x 的函数，则：

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

If y = u/v, then:

若 y = u/v，则：

dy/dx = (v du/dx − u dv/dx) / v²

The chain rule is used for composite functions. If y = f(g(x)), then dy/dx = f'(g(x)) g'(x). In Leibniz notation, this is often written as dy/dx = dy/du × du/dx.

链式法则用于复合函数。若 y = f(g(x))，则 dy/dx = f'(g(x)) g'(x)。在莱布尼茨记号中，这通常写成 dy/dx = dy/du × du/dx。

4. Derivatives of Trigonometric, Exponential and Logarithmic Functions | 三角函数、指数与对数函数的导数

Edexcel expects you to differentiate combinations of trigonometric, exponential and logarithmic functions accurately. The basic results must be combined with the chain rule when the input is a function of x.

Edexcel 要求你能够准确地对三角函数、指数函数和对数函数的组合求导。当输入是 x 的函数时，基本结果必须与链式法则结合使用。

d/dx [sin(kx)] = k cos(kx), d/dx [cos(kx)] = −k sin(kx), d/dx [eᵏˣ] = k eᵏˣ

For the natural logarithm, d/dx [ln(kx)] = 1/x. This is because ln(kx) = ln k + ln x and ln k is constant. For tan(kx), the derivative is k sec²(kx).

对于自然对数，d/dx [ln(kx)] = 1/x。这是因为 ln(kx) = ln k + ln x，而 ln k 是常数。对于 tan(kx)，其导数为 k sec²(kx)。

These results are especially common in questions on rates of change, stationary points and differential equations. Always check whether the chain rule is needed.

这些结果在变化率、驻点和微分方程问题中尤其常见。一定要检查是否需要使用链式法则。

5. Implicit Differentiation | 隐函数求导

When y is not given explicitly as a function of x, you can differentiate both sides of the equation with respect to x. Treat y as a function of x and differentiate any y term using the chain rule.

当 y 没有明确表示为 x 的函数时，你可以对方程两边同时关于 x 求导。将 y 视为 x 的函数，并对任何含 y 的项使用链式法则求导。

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

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

Implicit differentiation is essential for curves that are not functions, such as circles and some algebraic curves. It is also used to find gradients at points where explicit rearrangement is hard.

隐函数求导对于非函数曲线（如圆和某些代数曲线）至关重要。它也用于在难以显式重排的点处求梯度。

6. Parametric Differentiation | 参数方程求导

If x and y are both given as functions of a parameter t, the gradient dy/dx is found by dividing dy/dt by dx/dt. You must be confident with differentiating parametric equations.

如果 x 和 y 都表示为参数 t 的函数，则梯度 dy/dx 可通过 dy/dt 除以 dx/dt 得到。你必须熟练掌握参数方程求导。

dy/dx = (dy/dt) / (dx/dt)

The second derivative d²y/dx² is obtained by differentiating dy/dx with respect to t and then dividing by dx/dt. It is not simply the ratio of second derivatives.

二阶导数 d²y/dx² 通过先对 dy/dx 关于 t 求导，再除以 dx/dt 得到。它不是简单的二阶导数之比。

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

Parametric differentiation is common in questions involving curves defined by trigonometric or polynomial parametric equations. Always simplify using trigonometric identities where possible.

参数方程求导常见于由三角函数或多项式参数方程定义的曲线问题中。在可能的情况下，始终使用三角恒等式进行化简。

7. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative f”(x) measures the rate of change of the gradient. It tells you whether a curve is concave up or concave down at a given point.

二阶导数 f”(x) 衡量梯度的变化率。它告诉你曲线在给定点是凹向上还是凹向下。

If f”(x) > 0, the curve is concave up; if f”(x) < 0, it is concave down. A point where the concertina changes sign is called a point of inflection.

若 f”(x) > 0，曲线凹向上；若 f”(x) < 0，曲线凹向下。凹凸性改变符号的点称为拐点。

The second derivative is also used to classify stationary points. A positive second derivative indicates a local minimum, while a negative second derivative indicates a local maximum.

二阶导数也用于判断驻点类型。二阶导数为正表示局部极小值，二阶导数为负表示局部极大值。

8. Applications: Tangents and Normals | 应用：切线与法线

The derivative at a point gives the gradient of the tangent to the curve at that point. The normal is perpendicular to the tangent, so its gradient is the negative reciprocal.

函数在某点的导数给出曲线在该点切线的斜率。法线垂直于切线，因此其斜率为切线斜率的负倒数。

m_tangent = dy/dx at the point, m_normal = −1 / m_tangent

Once you have the gradient, use the straight-line equation y − y₁ = m(x − x₁) to write the equation of the tangent or normal.

一旦得到斜率，就使用直线方程 y − y₁ = m(x − x₁) 写出切线或法线的方程。

Always calculate the y-coordinate of the point before finding the line equation. Many candidates lose marks by using an incomplete point.

在求直线方程之前，始终先计算该点的 y 坐标。许多考生因使用不完整的点而失分。

9. Applications: Stationary Points and Optimisation | 应用：驻点与最优化

Stationary points occur where dy/dx = 0. You must be able to find them, classify them using the second derivative or a gradient table, and interpret them in context.

驻点出现在 dy/dx = 0 处。你必须能够找到它们，使用二阶导数或梯度表判断其类型，并在实际情境中解释其含义。

To classify a stationary point at x = a:

判断 x = a 处驻点类型的方法：

If f”(a) > 0, the point is a local minimum.
If f”(a) < 0, the point is a local maximum.
If f”(a) = 0, use a gradient table to check the sign change of f'(x).

若 f”(a) > 0，该点为局部极小值。
若 f”(a) < 0，该点为局部极大值。
若 f”(a) = 0，使用梯度表检查 f'(x) 的符号变化。

Optimisation questions ask you to model a real situation with a function and then use differentiation to find maximum or minimum values, such as maximum volume or minimum surface area.

最优化问题要求你用函数对实际情况建模，然后利用微分求最大值或最小值，例如最大体积或最小表面积。

10. Connected Rates of Change | 相关变化率

In connected rates of change problems, several variables depend on time. You use the chain rule to relate their rates of change. This is a very common Edexcel exam topic.

在相关变化率问题中，多个变量都随时间变化。你使用链式法则将它们的变率联系起来。这是 Edexcel 考试中非常常见的主题。

dV/dt = dV/dr × dr/dt

For example, if the volume of a sphere is V = 4/3 πr³, then dV/dr = 4πr². If you know dr/dt, you can find dV/dt at a particular radius.

例如，若球体体积为 V = 4/3 πr³，则 dV/dr = 4πr²。如果你知道 dr/dt，就可以求出特定半径下的 dV/dt。

Always identify the given rate and the required rate, then choose a chain of derivatives that connects them. Substitute values only after differentiating.

始终先识别已知变化率和所求变化率，然后选择连接它们的导数链。只在求导之后代入数值。

11. Exam Tips and Common Mistakes | 考试技巧与常见错误

Differentiation questions in Edexcel often combine several rules in one expression. Work step by step and do not rush the algebra. The most common errors include forgetting the chain rule, misapplying the quotient rule, and losing negative signs.

Edexcel 的微分题常在一个表达式中综合多个法则。要按步骤进行，不要急于完成代数运算。最常见的错误包括忘记链式法则、误用除法法则以及遗漏负号。

Before differentiating, rewrite roots and fractions as powers. After differentiating, simplify factorised answers where possible, but do not waste time on unnecessary expansion.

求导前，将根式和分式改写为幂形式。求导后，尽可能对答案进行因式化简，但不要在多余的展开上浪费时间。

In applied problems, always show the derivative you are using and state the value you are substituting. Use exact values such as π and √3 unless the question asks for decimals.

在应用题中，始终展示你使用的导数，并说明你代入的数值。除非题目要求小数，否则使用 π、√3 等精确值。

Finally, revise the standard derivatives table regularly and practise timed past-paper questions so that the rules become automatic under exam pressure.

最后，定期复习标准导数表，并在限时条件下练习历年真题，使这些法则在考试压力下也能自动化运用。

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Differentiation: Rules, Techniques and Exam Applications | 微分：法则、技巧与考试应用