Differentiation Revision for IB & CIE Maths | IB CIE 数学：微分考点精讲

📚 Differentiation Revision for IB & CIE Maths | IB CIE 数学：微分考点精讲

Differentiation is one of the foundational pillars of calculus, central to both IB and CIE A-Level Mathematics. It provides the tools to analyse rates of change, gradients of curves, and the behaviour of functions. Mastery of differentiation techniques and their applications is essential for success in examinations. This article breaks down the key concepts, rules, and problem-solving strategies you need to know, covering everything from first principles to implicit differentiation and optimisation problems.

微分是微积分的核心基石之一，对于 IB 和 CIE A-Level 数学都至关重要。它提供了分析变化率、曲线斜率以及函数行为的工具。掌握微分技巧及其应用是考试成功的关键。本文分解了你需要掌握的关键概念、法则和解题策略，涵盖从第一原理到隐函数微分和优化问题的所有内容。

1. The Gradient Function and First Principles | 斜率函数与第一原理

The derivative of a function f(x) at a point x = a is defined as the limit of the difference quotient: f'(a) = lim_h→0 [f(a+h) – f(a)] / h, provided the limit exists. This process, known as differentiation from first principles, gives the gradient of the tangent to the curve y = f(x) at that point.

函数 f(x) 在 x = a 处的导数定义为差商的极限：f'(a) = lim_h→0 [f(a+h) – f(a)] / h，前提是该极限存在。这个过程称为从第一原理求导，它给出了曲线 y = f(x) 在该点处切线的斜率。

For the general derivative function, we write f'(x) = lim_h→0 [f(x+h) – f(x)] / h. In examinations you may be asked to derive basic results using this limit definition. For example, for f(x) = x² we obtain f'(x) = 2x.

对于一般的导函数，我们写作 f'(x) = lim_h→0 [f(x+h) – f(x)] / h。在考试中，你可能会被要求使用这个极限定义推导基本结果。例如，对于 f(x) = x²，我们得到 f'(x) = 2x。

2. Basic Rules and Notation | 基本法则与符号

The derivative can be denoted in several ways: f'(x), dy/dx, y’, or d/dx [f(x)]. Each notation is widely used, and you must be comfortable switching between them. The power rule states that for f(x) = xⁿ, f'(x) = nxⁿ⁻¹, where n is any real number.

导数可以用多种方式表示：f'(x)、dy/dx、y’ 或 d/dx [f(x)]。每种符号都被广泛使用，你必须能熟练切换。幂法则指出，对于 f(x) = xⁿ，f'(x) = nxⁿ⁻¹，其中 n 为任意实数。

Other fundamental rules include the constant multiple rule: d/dx [c·f(x)] = c·f'(x), and the sum/difference rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). These allow differentiation of polynomial functions term by term.

其他基本法则包括常数倍法则：d/dx [c·f(x)] = c·f'(x)，以及和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。这些法则允许逐项对多项式函数求导。

3. Product and Quotient Rules | 乘积法则与商法则

For products of two functions, use the product rule: if y = u(x)v(x), then dy/dx = u’v + uv’. The order does not matter, but you must keep the pattern consistent. A common pitfall is forgetting to apply the chain rule when one of the functions is composite.

对于两个函数的乘积，使用乘积法则：如果 y = u(x)v(x)，那么 dy/dx = u’v + uv’。顺序无关紧要，但必须保持模式一致。一个常见的错误是当其中一个函数是复合函数时，忘记应用链式法则。

The quotient rule is used when one function divides another: if y = u(x) / v(x), then dy/dx = (u’v – uv’) / v². Note the minus sign in the numerator and the square of the denominator v(x). It is helpful to remember the mnemonic ‘low d-high minus high d-low over low squared’.

当一个函数除以另一个函数时，使用商法则：如果 y = u(x) / v(x)，那么 dy/dx = (u’v – uv’) / v²。注意分子中的减号以及分母 v(x) 的平方。记住口诀 “分母乘分子的导数减去分子乘分母的导数，整体除以分母的平方” 会很有帮助。

4. The Chain Rule | 链式法则

The chain rule is used to differentiate composite functions: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x). In Leibniz notation, this is written as dy/dx = dy/du · du/dx, where u = g(x). This rule is essential for differentiating powers of functions, exponentials, logarithms, and trigonometric functions with linear or more complex arguments.

链式法则用于求复合函数的导数：如果 y = f(g(x))，那么 dy/dx = f'(g(x)) · g'(x)。在莱布尼茨符号中，写作 dy/dx = dy/du · du/dx，其中 u = g(x)。这个法则对于求导函数的幂、指数、对数以及带有线性或更复杂参数的三角函数至关重要。

For example, to differentiate y = (3x² + 5)⁴, set u = 3x² + 5, then y = u⁴, so dy/dx = 4u³ · du/dx = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³. Without the chain rule, such differentiation would be extremely cumbersome.

例如，要对 y = (3x² + 5)⁴ 求导，设 u = 3x² + 5，则 y = u⁴，因此 dy/dx = 4u³ · du/dx = 4(3x² + 5)³ · 6x = 24x(3x² + 5)³。没有链式法则，这类求导将极其繁琐。

5. Derivatives of Trigonometric Functions | 三角函数的导数

You must memorise the standard derivatives of the six trigonometric functions. The derivatives of sin x, cos x and tan x are essential for any calculus exam. Familiarity with the reciprocal functions (sec, csc, cot) is also required, particularly in IB Higher Level and CIE Further Mathematics.

你必须记住六种三角函数的标准导数。sin x、cos x 和 tan x 的导数是任何微积分考试的基础。熟悉倒数函数（sec、csc、cot）也是必需的，尤其是在 IB 更高级别和 CIE 进阶数学中。

Function f(x)	Derivative f'(x)
sin x	cos x
cos x	-sin x
tan x	sec² x
cot x	-csc² x
sec x	sec x tan x
csc x	-csc x cot x

When the argument is more complex than x, the chain rule applies. For instance, d/dx [sin(ax + b)] = a·cos(ax + b). Being able to apply these rules quickly and accurately will save time in applications such as solving differential equations or analysing oscillatory motion.

当参数比 x 更复杂时，需要应用链式法则。例如，d/dx [sin(ax + b)] = a·cos(ax + b)。能够快速准确地应用这些规则，将在解微分方程或分析振动运动等应用中节省时间。

6. Exponential and Logarithmic Differentiation | 指数与对数函数的微分

The derivative of eˣ is simply eˣ, making it unique. For a general exponential function aˣ, the derivative is aˣ ln a. The natural logarithm function ln x has derivative 1/x. For ln[f(x)], the chain rule gives d/dx ln[f(x)] = f'(x)/f(x). This is a crucial technique for integrating rational functions and for logarithmic differentiation.

eˣ 的导数就是 eˣ，这使它独一无二。对于一般指数函数 aˣ，导数为 aˣ ln a。自然对数函数 ln x 的导数为 1/x。对于 ln[f(x)]，链式法则给出 d/dx ln[f(x)] = f'(x)/f(x)。这是积分有理函数和对数求导法的关键技巧。

Logarithmic differentiation is particularly useful when differentiating functions of the form y = [f(x)]^g(x). By taking natural logs of both sides, ln y = g(x)·ln[f(x)], and then differentiating implicitly, you can handle otherwise intractable derivatives.

对数求导法在求导形如 y = [f(x)]^g(x) 的函数时特别有用。通过对两边取自然对数，ln y = g(x)·ln[f(x)]，然后隐式求导，你可以处理原本难以处理的导数。

7. Implicit Differentiation | 隐函数求导

When a relation between x and y is given implicitly, such as x² + xy + y² = 7, differentiate both sides with respect to x, treating y as a function of x. Each time you differentiate a term containing y, multiply by dy/dx due to the chain rule. Then solve for dy/dx algebraically.

当 x 和 y 之间的关系以隐式形式给出时，如 x² + xy + y² = 7，对两边关于 x 求导，将 y 视为 x 的函数。每次你对含有 y 的项求导时，都要根据链式法则乘以 dy/dx。然后通过代数方法求解 dy/dx。

For the example above: 2x + (y + x·dy/dx) + 2y·dy/dx = 0. Collect terms: (x + 2y)·dy/dx = -(2x + y), so dy/dx = -(2x + y)/(x + 2y). Implicit differentiation is essential for finding tangents to curves defined by equations that cannot be easily expressed as y = f(x).

对于上例：2x + (y + x·dy/dx) + 2y·dy/dx = 0。合并同类项：(x + 2y)·dy/dx = -(2x + y)，因此 dy/dx = -(2x + y)/(x + 2y)。隐函数求导对于找到由无法轻易表示为 y = f(x) 的方程定义的曲线的切线至关重要。

8. Parametric Differentiation | 参数微分

When a curve is defined parametrically by x = f(t), y = g(t), the derivative dy/dx is given by (dy/dt) / (dx/dt), provided dx/dt ≠ 0. The second derivative d²y/dx² is found by differentiating dy/dx with respect to x: d²y/dx² = d/dt (dy/dx) / (dx/dt).

当曲线由参数方程 x = f(t), y = g(t) 定义时，导数 dy/dx 由 (dy/dt) / (dx/dt) 给出，前提是 dx/dt ≠ 0。二阶导数 d²y/dx² 通过 dy/dx 再对 x 求导得到：d²y/dx² = d/dt (dy/dx) / (dx/dt)。

This technique is heavily tested in CIE Mechanics (M1/M2) and in IB Analysis & Approaches. Common applications include finding equations of tangents and normals at a given parameter value, and determining the nature of stationary points for parametric curves.

这项技巧在 CIE 力学 (M1/M2) 和 IB 分析与方法中考查频繁。常见应用包括在给定参数值处求切线和法线方程，以及判断参数曲线驻点的性质。

9. Higher-Order Derivatives | 高阶导数

The second derivative, f”(x) or d²y/dx², measures the rate of change of the gradient, giving information about the concavity of the function. Higher-order derivatives (f”'(x), f⁽⁴⁾(x), etc.) appear in Taylor series expansions, differential equations, and in problems involving displacement, velocity, acceleration, and jerk in kinematics.

二阶导数 f”(x) 或 d²y/dx² 衡量斜率的变化率，提供了函数凹凸性的信息。高阶导数（f”'(x), f⁽⁴⁾(x) 等）出现在泰勒级数展开、微分方程以及运动学中涉及位移、速度、加速度和加加速度的问题中。

In CIE Mechanics, if displacement s = f(t), then velocity v = ds/dt, acceleration a = dv/dt = d²s/dt². Interpreting these derivatives graphically and algebraically is a core skill.

在 CIE 力学中，如果位移 s = f(t)，则速度 v = ds/dt，加速度 a = dv/dt = d²s/dt²。从图形和代数上解读这些导数是一项核心技能。

10. Applications: Tangents, Normals, and Gradient | 应用：切线、法线与斜率

The derivative at a point gives the gradient m of the tangent line to the curve at that point. The equation of the tangent is then y – y₁ = m(x – x₁). The normal line is perpendicular to the tangent, so its gradient is -1/m (provided m ≠ 0), and its equation is y – y₁ = (-1/m)(x – x₁).

某点处的导数给出了曲线在该点处切线的斜率 m。切线方程则为 y – y₁ = m(x – x₁)。法线垂直于切线，因此其斜率为 -1/m（假设 m ≠ 0），其方程为 y – y₁ = (-1/m)(x – x₁)。

Problems often ask for the equations of tangents and normals at specific points, the coordinates of points where the tangent is horizontal or parallel to a given line, and the intersection points of tangents with axes.

题目经常要求求特定点处的切线和法线方程、切线水平或平行于给定直线的点的坐标，以及切线与坐标轴的交点。

11. Stationary Points and Curve Sketching | 驻点与曲线绘制

Stationary points occur where the first derivative dy/dx = 0. They can be classified using either the first derivative test (sign change of dy/dx) or the second derivative test (sign of d²y/dx²). Local maxima, local minima, and points of inflection (horizontal or non-horizontal) are vital for sketching accurate graphs.

驻点出现在一阶导数 dy/dx = 0 处。它们可以使用一阶导数判别法（dy/dx 的符号变化）或二阶导数判别法（d²y/dx² 的符号）进行分类。局部极大值、局部极小值和拐点（水平或非水平）对于绘制精确的图形至关重要。

If d²y/dx² > 0 at a stationary point, it is a local minimum. | 如果在驻点处 d²y/dx² > 0，则为局部极小值。
If d²y/dx² < 0, it is a local maximum. | 如果 d²y/dx² < 0，则为局部极大值。
If d²y/dx² = 0, the test is inconclusive; use the first derivative test. | 如果 d²y/dx² = 0，该判别法无法确定；使用一阶导数判别法。

Additionally, the sign of d²y/dx² indicates concavity: positive means concave up, negative means concave down. Points of inflection are where concavity changes.

此外，d²y/dx² 的符号表示凹凸性：正表示凹向上，负表示凹向下。拐点是凹凸性发生变化的地方。

12. Optimisation and Related Rates | 优化与相关变化率

Optimisation involves finding maximum or minimum values of a function within a domain, often motivated by physical constraints (e.g., minimising surface area for a fixed volume). The process involves modelling the quantity to optimise in terms of one variable, differentiating, finding stationary points, and verifying the nature by a derivative test. Always check endpoints in closed interval problems.

优化涉及在一个定义域内找到函数的最大值或最小值，通常由物理约束驱动（例如，在固定体积下最小化表面积）。过程包括将待优化量建模为一个变量的函数，求导，找到驻点，并通过导数判别法验证性质。在闭区间问题中，始终检查端点。

Related rates problems use the chain rule to relate rates of change of two or more variables with respect to time. For instance, if a circular ripple expands, the rate of change of area dA/dt is related to the rate of change of radius dr/dt by dA/dt = dA/dr · dr/dt = 2πr · dr/dt. Such problems require careful differentiation and substitution of given values.

相关变化率问题利用链式法则将两个或多个变量关于时间的变化率联系起来。例如，如果一个圆形波纹扩展，面积变化率 dA/dt 与半径变化率 dr/dt 通过 dA/dt = dA/dr · dr/dt = 2πr · dr/dt 联系起来。这类问题需要仔细求导并代入给定数值。

Both IB and CIE exams frequently include contextual problem-solving questions featuring optimisation in geometry, economics, and kinematics. Practising a variety of applied differentiation questions is essential to build confidence.

IB 和 CIE 考试都经常包含情境化问题解决题，涉及几何、经济学和运动学中的优化。练习各种应用微分问题对于建立信心至关重要。

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Differentiation Revision for IB & CIE Maths | IB CIE 数学：微分 考点精讲