📚 Review Set 9B: Differentiation | 复习题集 9B：微分
This review set covers the core differentiation topics typically found in Chapter 9 of the IB Mathematics: Analysis and Approaches HL course. Mastering these concepts is essential for success in calculus and related applications throughout the Diploma Programme. The following guide summarises key rules, techniques, and common pitfalls to help you reinforce your understanding and perform confidently in assessments.
本复习题集涵盖 IB 数学分析与方法高阶课程第 9 章中常见的微分核心主题。掌握这些概念对于在整个文凭课程中成功掌握微积分及其应用至关重要。以下指南总结了关键法则、技巧和常见错误，帮助你巩固理解并在评估中自信应对。
1. Definition of the Derivative | 导数的定义
The derivative of a function f(x) at a point x = a is defined as the limit f'(a) = limh→0 (f(a + h) – f the function or the slope of the tangent line at that point. For a general x, we write f'(x) = limh→0 (f(x + h) – f(x)) / h.
函数 f(x) 在 x = a 处的导数定义为极限 f'(a) = limh→0 (f(a + h) – f(a)) / h，前提是该极限存在。这一基本概念衡量了函数的瞬时变化率或该点处切线的斜率。对于一般的 x，我们写作 f'(x) = limh→0 (f(x + h) – f(x)) / h。
Understanding the limit definition is critical because it forms the basis for all differentiation rules. It also appears in first-principles questions on IB exams, where you may be asked to differentiate simple functions like x² or 1/x directly from the definition.
理解极限定义至关重要，因为它是所有微分法则的基础。它也出现在 IB 考试的“从第一原理出发”的问题中，你可能需要直接从定义出发对简单函数如 x² 或 1/x 求导。
2. Basic Differentiation Rules | 基本求导法则
The power rule states that if f(x) = xⁿ, then f' le of a function is the constant times the derivative of the function: d/dx [c·f(x)] = c·f'(x).
幂法则指出，若 f(x) = xⁿ，则 f'(x) = nxⁿ⁻¹ 对任意实数指数 n 成立。常数的导数为零，且常数与函数乘积的导数等于该常数乘以函数的导数：d/dx [c·f(x)] = c·f'(x)。
he sum and difference rules allow us to differentiate term by term: d/dx [f(x) ± g(x)] = f'(x) ± g' power rule, these enable rapid differentiation of polynomials. For example, if f(x) = 3x⁴ – 5x² + 2x – 7, then f'(x) = 12x³ – 10x + 2.
和差法则允许我们逐项求导：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。结合幂法则，这些规则可以快速对多项式求导。例如，若 f(x) = 3x⁴ – 5x² + 2x – 7，则 f'(x) = 12x³ – 10x + 2。
3. The Chain Rule | 链式法则
u = g(x), then dy/dx = dy/du × du/dx. In function notation, the derivative of f(g(x)) is f'(g(x)) × g'x). This rule is indispensable when dealing with functions such as sin(2x), (3x²+1)⁵, or eˣ².
链式法则用于求复合函数的导数。若 y = f(u) 且 u = g(x)，则 dy/dx = dy/du × du/dx。用函数符号表示为，f(g(x)) 的导数是 f'(g(x)) × g'(x)。在处理诸如 sin(2x)、(3x²+1)⁵ 或 eˣ² 等函数时，这一法则不可或缺。
A common strategy is to identify the "inner" function and the "outer" function. For instance, for y = (5x³ – x)⁸, the outer function is u⁸ and the inner is 5x³ – x. The derivative becomes 8(5x³ – x)⁷ × (15x² – 1). Always multiply by the derivative of the inner function.
常见的策略是识别“内层”函数和“外层”函数。例如，对于 y = (5x³ – x)⁸，外层函数是 u⁸，内层是 5x³ – x。导数即为 8(5x³ – x)⁷ × (15x² – 1)。务必乘以内层函数的导数。
| Chain Rule Form | Example |
| dy/dx = f'(u) · u' | y = sin(x²) → dy/dx = cos(x²) · 2x |
4. Product and Quotient Rules | 乘积法则与商法则
When differentiating the product of two functions, use the product rule: (f·g)' = f '·g + f·g'. For y = x²·sin x, the derivative is y' = 2x·sin x + x²·cos x. Order does not matter as long as both terms appear.
对两个函数的乘积求导时，使用乘积法则：(f·g)' = f '·g + f·g'。对于 y = x²·sin x，导数为 y' = 2x·sin x + x²·cos x。只要两项都出现，顺序无关紧要。
The quotient rule handles division: if y = f/g, then y' = (f '·g – f·g') / g². A typical memory aid is "low d high minus high d low, square the bottom and away we go". For example, the derivative of x/cos x is (1·cos x – x·(– x)) / cos² x = (cos x + x sin x) / cos² x. Always be careful with signs when subtracting.
商法则处理除法：若 y = f/g，则 y' = (f '·g – f·g') / g²。典型的记忆方法是“分母乘分子的导数减分子乘分母的导数，再除以分母的平方”。例如，x/cos x 的导数为 (1·cos x – x·(–sin x)) / cos² x = (cos x + x sin x) / cos² x。相减时务必注意符号。
+1)¹⁰ requires the product rule, where the derivative of the second factor uses the chain rule: 10(2x+1)⁹·2. Such layered differentiation appears frequently.
在 IB 问题中，你可能需要将这些法则与链式法则结合使用。例如，对 y = x·(2x+1)¹⁰ 求导需要乘积法则，而第二个因子的导数需用到链式法则：10(2x+1)⁹·2。此类分层求导经常出现。
5. Derivatives of Trigonometric Functions | 三角函数的导数
st be memorised, with special attention to the signs. d/dx (sin x) = cos x, d/dx (cos x) = – tive of tan x is sec² x, which can be derived from the quotient rule: tan x = sin x/cos x.
六个标准三角函数的导数必须牢记，尤其要注意符号。d/dx (sin x) = cos x，d/dx (cos x) = –sin x。tan x 的导数为 sec² x，这可由商法则推导得出：tan x = sin x/cos x。
- d/dx (sec x) = sec x tan x
- d/dx (csc x) = –csc x cot x
- d/dx (cot x) = –csc² x
d/dx [sin(4x)] = 4 cos(4x). Similarly, d/dx [cos² x] means (cos x)², yielding 2 cos x · (–sin x) = –2 sin x cos x = –sin(2x). Recognising these patterns saves time.
当自变量是 x 的函数时，需应用链式法则。例如，d/dx [sin(4x)] = 4 cos(4x)。类似地，d/dx [cos² x] 即 (cos x)²，得到 2 cos x · (–sin x) = –2 sin x cos x = –sin(2x)。识别这些模式可节省时间。
6. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数
The derivative of eˣ is eˣ , making it one of the most important functions in calculus. For a general base a, d/dx (aˣ) = aˣ ln a. When the exponent is a function, use the chain rule: d/dx (eᵘ) = eᵘ · u'.
eˣ 的导数是 eˣ 本身，这使其成为微积分中最重要的函数之一。对于一般底数 a，d/dx (aˣ) = aˣ ln a。当指数为函数时，使用链式法则：d/dx (eᵘ) = eᵘ · u'。
tions, d/dx (ln x) = 1/x for x > 0. If u is a positive function of x, then d/dx (ln u) = u' ith other bases, use d/dx (logₐ x) = 1/(x ln a) or first apply the change-of-base formula.
对于自然对数函数，d/dx (ln x) = 1/x（x > 0）。若 u 是 x 的正函数，则 d/dx (ln u) = u'/u。对于其他底数的对数，可使用 d/dx (logₐ x) = 1/(x ln a) 或先用换底公式。
thmic differentiation to handle complicated products. For instance, to differentiate y = xˣ, take natural logarithms: ln y = x ln x, then differentiate implicitly to obtain y' = xˣ (ln x + 1).
指数与对数微分的结合常出现在隐函数求导中，或使用对数微分法处理复杂乘积时。例如，要求 y = xˣ 的导数，可对两边取自然对数：ln y = x ln x，然后隐函数求导得 y' = xˣ (ln x + 1)。
7. Implicit Differentiation | 隐函数求导
we differentiate both sides with respect to x, treating y as a function of x. Each time we differentiate a term in y, we multiply by dy/dx due to the chain rule.
当 x 与 y 的关系以隐式给出时（例如 x² + y² = 25），我们对两边关于 x 求导，并将 y 视为 x 的函数。每次对含 y 的项求导时，都要根据链式法则乘上 dy/dx。
For the circle equation, differentiating gives 2x + 2y(dy/dx) = 0, so dy/dx = – f(x). It also appears in optimisation problems where variables are linked by an equation.
对于圆的方程，求导得 2x + 2y(dy/dx) = 0，因此 dy/dx = –x/y。这一技巧对于求不能简单写成 y = f(x) 的曲线的切线至关重要。它也出现在变量由方程关联的最优化问题中。
rcsin x. Start with sin y = x, differentiate to cos y (dy/dx) = 1, so dy/dx = 1/cos y = 1/√1 – x². This demonstrates how inverse trigonometric derivatives are derived.
隐函数求导使我们能处理诸如 y = arcsin x 的函数。从 sin y = x 出发，求导得 cos y (dy/dx) = 1，因此 dy/dx = 1/cos y = 1/√1 – x²。这展示了反三角函数导数是如何推导出来的。
8. Higher-Order Derivatives | 高阶导数
The second derivative, denoted f''(x) or d²y/dx², is the derivative of the first derivative. It measures the rate of change of the gradient and determines concavity. A positive second derivative indicates the function is concave up (like a cup), while a negative value means concave down.
二阶导数记作 f''(x) 或 d²y/dx²，是一阶导数的导数。它衡量了斜率的变化率，并确定函数的凹凸性。正的二阶导数表示函数开口向上（凹向上），负值则表示开口向下。
quired in Taylor polynomials or in repeated differentiation tasks. For instance, finding f''' (x) for f(x) = x eˣ nvolves multiple applications of the product rule. In IB, you might be asked to evaluate f'' (a) at a given point to classify stationary points.
高阶导数（三阶、四阶......）可能在泰勒多项式或多次求导任务中需要。例如，对 f(x) = x eˣ 求 f''' (x) 需多次应用乘积法则。在 IB 中，可能要求你在给定点计算 f'' (a) 以对驻点进行分类。
Notation can vary: f' (Lagrange), dy/dx (Leibniz), or ẏ (Newton' t notation) in kinematics. In IB kinematics problems, displacement s(t), velocity v(t) = s'(t), and acceleration a(t) = v'(t) = s''(t) exemplify higher-order derivatives in context.
记法可能有所不同：f'（拉格朗日记法）、dy/dx（莱布尼茨记法），或在运动学中使用 ẏ（牛顿点记法）。在 IB 运动学问题中，位移 s(t)、速度 v(t) = s'(t) 和加速度 a(t) = v'(t) = s''(t) 正是高阶导数在情境中的体现。
9. Tangents and Normals | 切线与法线
The equation of the tangent line to y = f(x) at x = a is y – f(a) = f'(a)(x – a). The normal line is perpendicular to the tangent and has slope –1/f'(a), provided f'(a) ≠ 0. Its equation is y – f(a) = –1/f'(a) (x – a).
曲线 y = f(x) 在 x = a 处的切线方程为 y – f(a) = f'(a)(x – a)。法线垂直于切线，斜率为 –1/f'(a)（假设 f'(a) ≠ 0）。其方程为 y – f(a) = –1/f'(a) (x – a)。
Finding the point where a tangent has a specific slope involves setting f'x) equal to that slope and solving for x. For example, to find where the tangent to y = x³ – 3x is horizontal, set f'(x) = 3x² – 3 = 0, giving x = ±1. Horizontal tangents correspond to stationary points.
寻找切线具有特定斜率的点，需要令 f'(x) 等于该斜率并解出 x。例如，要找到 y = x³ – 3x 的水平切线位置，令 f'(x) = 3x² – 3 = 0，解得 x = ±1。水平切线对应驻点。
IB questions often ask for the equation of a normal that passes through a given point not on the curve, requiring careful substitution and algebraic solving. Always check that the point of tangency satisfies both the curve and the tangent/normal equation.
IB 问题常要求求出通过不在曲线上某给定点的法线方程，这需要仔细代入和代数求解。始终检查切点同时满足曲线方程和切线/法线方程。
10. Optimisation Problems | 最优化问题
xpress it as a function of one variable using given constraints, find stationary points by solving f'(x) = 0, and determine the nature using the second derivative or sign change of f'(x).
最优化涉及在给定定义域内寻找函数的最大值或最小值。步骤包括：确定要优化的变量，利用给定约束将其表示为单一变量的函数，通过解 f'(x) = 0 找到驻点，并用二阶导数或 f'(x) 的符号变化判断性质。
Common IB contexts: minimising surface area for a fixed volume (cylinders, boxes), maximising area enclosed by a fence, or minimising cost. Always verify endpoints of the domain if the function is defined on a closed interval, as global extrema may occur there.
常见的 IB 情境包括：固定体积下最小化表面积（圆柱体、盒子）、最大化围栏面积或最小化成本。若函数定义在闭区间上，务必验证端点，因为全局极值可能出现在端点处。
Using the second derivative test: if f''(c) > 0, the point is a local minimum; if f''(c) < 0, it is a local maximum. If f'' the test is inconclusive, and the first derivative test should be used. Providing a clear justification is essential for full marks.
使用二阶导数检验：若 f''(c) > 0，则该点为局部极小值；若 f''(c) < 0，则为局部极大值。若 f''(c) = 0，则检验无效，应使用一阶导数检验。提供清晰的理由对于获得满分至关重要。
11. Related Rates | 相关变化率
Related rates problems involve finding the rate at which one quantity changes with respect to time given the rate of change of a related quantity. The key is to write an equation linking the variables, then differentiate both sides with respect to time t using the chain rule implicitly.
相关变化率问题涉及已知一个相关量的时间变化率，求另一个量的时间变化率。关键在于写出变量之间的关联方程，然后利用链式法则隐式地对时间 t 两边求导。
For example, if a spherical balloon' the given rates and the required rate, substitute known values only after differentiation, and state the units clearly.
例如，若球形气球的半径 r 以恒定速率 dr/dt 增加，则体积的变化率为 dV/dt = 4πr² dr/dt。始终先识别已知变化率和需求变化率，求导后再代入已知数值，并清晰注明单位。
Typical IB scenarios include water leaking from a conical tank, a sliding ladder, or the distance between two moving cars. Drawing a diagram and writing a geometric relationship (Pythagoras, similar triangles, volume formulas) is the first step. Pay attention to signs: decreasing quantities have negative rates.
典型的 IB 场景包括圆锥水箱漏水、滑动的梯子或两辆移动汽车之间的距离。绘制示意图并写出几何关系（勾股定理、相似三角形、体积公式）是第一步。注意符号：减少的量速率为负值。
12. Summary & Common Pitfalls | 总结与常见错误
Mastering differentiation requires consistent practice and attention to detail. Memorise the basic derivative formulas for powers, exponentials, logs, and trigonometric functions. Combine the chain rule automatically whenever the argument is not a simple x. Use the product and quotient rules only when necessary; sometimes simplifying the expression first (e.g., expanding or splitting fractions) makes differentiation easier.
掌握微分需要持续练习和对细节的关注。熟记幂函数、指数、对数和三角函数的导数基本公式。当自变量不是简单的 x 时，自动结合链式法则。仅在必要时使用乘积和商法则；有时先化简表达式（如展开或拆分分式）会使求导更简单。
Common mistakes include: forgetting to multiply by the derivative of the inner function, misapplying the signs in trigonometric derivatives (especially cos and csc), incorrectly using the quotient rule when a constant multiple could be taken out, and algebra errors when simplifying answers. Always ask yourself if the derivative is reasonable in terms of expected degree or behaviour. Checking with a graphing calculator can help build intuition.
常见错误包括：忘记乘以内层函数的导数、三角函数导数符号使用错误（尤其是 cos 和 csc）、在可以提取常数倍数时误用商法则、化简答案时的代数错误。始终自问导数在预期次数或行为方面是否合理。使用图形计算器检查有助于培养直觉。
This Review Set 9B is designed to help you consolidate these skills. Work through each problem carefully, showing all steps of differentiation and fully simplifying your final answers. When stuck, revisit the relevant section in this guide. Consistent practice with a variety of functions—polynomials, rational, exponential, logarithmic, and trigonometric—will build your confidence and speed. Good luck!
本复习题集 9B 旨在帮助你巩固这些技能。仔细完成每道题目，展示微分的全部步骤，并充分化简最终答案。遇到困难时，重新阅读本指南相关章节。通过多样化函数（多项式、有理函数、指数、对数和三角函数）的持续练习，将提升你的信心和速度。祝你好运！
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