📚 IB Mathematics: Memorising & Using the Basic Derivatives Table | IB数学：基本导数表记忆与使用
Differentiation is one of the most powerful tools in IB Mathematics, appearing in both Analysis & Approaches (AA) and Applications & Interpretation (AI). Yet many students lose marks not because they cannot differentiate, but because they misremember the basic derivative table. This article will show you how to memorise the table effectively and apply it correctly in exam contexts.
微分是IB数学中最强大的工具之一，无论是在分析与方法（AA）还是应用与解释（AI）课程中都频繁出现。然而许多学生丢分的原因并非不会求导，而是记错了基本导数表。本文将向你展示如何高效记忆导数表，并在考试情境中正确使用。
1. The Definition of the Derivative | 导数的定义
= a is defined as the limit of the average rate of change as the interval approaches zero:
在记忆公式之前，你必须理解导数的本质。函数 f(x) 在 x = a 处的导数定义为当区间趋近于零时平均变化率的极限：
f'(a) = lim[h→0] [f(a + h) − f(a)] / h
This limit definition is the foundation of everything in differential calculus. In IB exams, you may occasionally be asked to differentiate from first principles, especially for simple polynomials.
这个极限定义是整个微分学的基础。在IB考试中，偶尔会要求你从第一原理出发求导，尤其是对简单多项式函数。
2. The Power Rule — Your Most Used Formula | 幂法则——最常用的公式
For any function of the form f(x) = xⁿ, where n is a real number, the derivative is:
对于形如 f(x) = xⁿ 的函数，其中 n 为实数，其导数为：
d/dx (xⁿ) = n·xⁿ⁻¹
This rule applies to positive powers, negative powers, and fractional powers alike.
该规则同样适用于正幂、负幂和分数幂。
- Example 1: d/dx (x⁵) = 5x⁴
- 例子1： d/dx (x⁵) = 5x⁴
- Example 2: d/dx (x⁻³) = −3x⁻⁴
- 例子2： d/dx (x⁻³) = −3x⁻⁴
- Example 3: d/dx (√x) = d/dx (x^½) = ½x−½ = 1/(2√x)
- 例子3： d/dx (√x) = d/dx (x^½) = ½x−½ = 1/(2√x)
A common IB trap is forgetting to rewrite radicals or reciprocal powers before differentiating.
一个常见的IB陷阱是忘记在求导之前将根式或倒数幂改写为指数形式。
3. Trigonometric Derivatives | 三角函数的导数
The derivatives of the six trigonometric functions form a core part of the IB formula booklet, but you must know how to use them fluently.
六个三角函数的导数是IB公式手册的核心部分，但你必须能够熟练使用它们。
| Function 函数 | Derivative 导数 |
| sin x | cos x |
| cos x | −sin x |
| tan x | sec² x |
| sec x | sec x · tan x |
| csc x | −csc x · cot x |
| cot x | −csc² x |
A useful memory device: the derivatives of functions starting with 'c' (cos, csc, cot) are all negative.
一个有用的记忆技巧：以字母'c'开头的函数（cos、csc、cot）的导数都带有负号。
4. Exponential and Logarithmic Derivatives | 指数与对数函数的导数
Exponential and logarithmic functions follow special differentiation rules that are easy to mix up.
指数函数和对数函数遵循特殊的求导规则，容易混淆。
d/dx (eˣ) = eˣ
d/dx (aˣ) = aˣ · ln a (a > 0, a ≠ 1)
d/dx (ln x) = 1/x (x > 0)
d/dx (logₐ x) = 1/(x · ln a) (x > 0)
The fact that eˣ is its own derivative is what makes the natural exponential function so special in calculus. For logarithms, the natural logarithm (ln) has the cleanest derivative.
eˣ 的导数就是自身，这一特性使得自然指数函数在微积分中如此特殊。对于对数而言，自然对数 ln 拥有最简洁的导数形式。
5. The Chain Rule — Beyond the Basic Table | 链式法则——超越基本表
The basic derivative table only gives derivatives of simple functions. When you have a composite function like sin(x²) or e3x, you must apply the chain rule.
基本导数表只给出简单函数的导数。当你面对像 sin(x²) 或 e3x 这样的复合函数时，必须使用链式法则。
d/dx [f(g(x))] = f'(g(x)) · g'(x)
This is often summarised as "differentiate the outside, keep the inside, multiply by the derivative of the inside."
这通常被概括为"对外层函数求导，保持内层不变，再乘以内层函数的导数"。
- Example: d/dx [sin(x²)] = cos(x²) · 2x
- 例子： d/dx [sin(x²)] = cos(x²) · 2x
- Example: d/dx [e3x] = e3x · 3
- 例子： d/dx [e3x] = e3x · 3
6. The Product and Quotient Rules | 乘法法则与除法法则
When two functions are multiplied or divided, you cannot simply differentiate each part separately.
当两个函数相乘或相除时，你不能简单地对各部分分别求导。
Product Rule: d/dx [u·v] = u·v' + v·u'
Quotient Rule: d/dx [u/v] = (v·u' − u·v') / v²
A common mnemonic for the quotient rule is "low d-high minus high d-low, over low squared."
商法则的常用口诀是"上导下不导减下导上不导，除以分母的平方"。
| Function 函数 | Use 使用 |
| x²·ln x | Product Rule 乘法法则 |
| x / (x²+1) | Quotient Rule 除法法则 |
| (x+1)⁵ | Chain Rule 链式法则 |
7. Common Mistakes to Avoid | 常见错误及避免方法
Even top IB students make predictable errors with the derivative table. Here are the most frequent pitfalls.
即使是顶尖的IB学生，也会在导数表上犯一些可预测的错误。以下是最高频的陷阱。
- Mistake 1: Writing d/dx (eˣ) = x·eˣ⁻¹ (using the power rule on eˣ). The power rule only applies when the variable is in the base, not the exponent.
- 错误1： 将 d/dx (eˣ) 误写为 x·eˣ⁻¹（对 eˣ 使用幂法则）。幂法则仅适用于变量在底数位置，而非指数位置。
- Mistake 2: Forgetting to apply the chain rule: d/dx (sin 2x) ≠ cos 2x. The correct answer is 2cos 2x.
- 错误2： 忘记使用链式法则：d/dx (sin 2x) ≠ cos 2x。正确答案是 2cos 2x。
- Mistake 3: Sign errors with negative powers: d/dx (x⁻²) = −2x⁻³, not −2x⁻¹.
- 错误3： 负幂的符号错误：d/dx (x⁻²) = −2x⁻³，而不是 −2x⁻¹。
- Mistake 4: Confusing ln x and log₁₀ x: the derivative of ln x is 1/x, but the derivative of log₁₀ x is 1/(x·ln 10).
- 错误4： 混淆 ln x 和 log₁₀ x：ln x 的导数为 1/x，而 log₁₀ x 的导数为 1/(x·ln 10)。
8. Memory Techniques for the Derivative Table | 导数表记忆技巧
Rote memorisation is not enough — you need meaningful patterns to retain these formulas under exam pressure.
死记硬背是不够的——你需要有意义的模式，才能在考试压力下牢记这些公式。
- Group by family: All trigonometric derivatives are interlinked: sin → cos → −sin → −cos → sin (a cycle).
- 按家族分组： 所有三角函数的导数都是相互关联的：sin → cos → −sin → −cos → sin（一个循环）。
- Associate with graphs: The slope of sin x at x = 0 is 1, matching cos 0 = 1. Visual connections reinforce memory.
- 与图像关联： sin x 在 x = 0 处的斜率为 1，恰好 cos 0 = 1。视觉联系能强化记忆。
- Mnemonics for signs: "Co-functions have negative derivatives" — cos, cot, csc all start with 'co' (or 'c') and carry a minus sign.
- 符号记忆法： "余函数（co-函数）的导数带负号"——cos、cot、csc 都以'co'（或'c'）开头，导数均带负号。
- Practice with flashcards: Write the function on one side and the derivative on the other. Test yourself weekly.
- 使用闪卡练习： 一面写函数，另一面写导数。每周自测一次。
9. Contextual Problems — IB Style Exam Questions | 情境题——IB风格考试题目
In IB exams, you will rarely be asked to "differentiate" in isolation. The derivative is usually embedded in a contextual problem involving tangent lines, rates of change, or optimisation.
在IB考试中，很少会孤立地要求你"求导"。导数通常嵌入在涉及切线、变化率或最优化的情境题中。
Example (AA-style question): ed in seconds. Find the velocity and acceleration functions, and determine when the particle is at rest.
示例（AA风格题目）： 质点的位置函数为 s(t) = t³ − 6t² + 9t，其中 t ≥ 0 以秒为单位。求速度函数和加速度函数，并确定质点何时静止。
Solution: v(t) = s'(t) = 3t² − 12t + 9; a(t) = v'(t) = 6t − 12
s 3(t² − 4t + 3) = 3(t − 1)(t − 3) = 0, so the particle is at rest at t = 1 s and t = 3 s.
令 v(t) = 0 得 3(t² − 4t + 3) = 3(t − 1)(t − 3) = 0，因此质点在 t = 1 s 和 t = 3 s 时静止。
10. The Derivative Table in Your IB Formula Booklet | IB公式手册中的导数表
For IB AA, the formula booklet (Formula Booklet AA SL/HL) includes a section on derivatives. However, you will not find every derivative — many must be derived using rules.
对于IB AA而言，公式手册（Formula Booklet AA SL/HL）包含导数部分。但你并不会在那里找到所有导数——许多导数需要利用法则推导。
| In Booklet 手册中 | Not in Booklet 手册中没有 |
| sin x, cos x, tan x | sec x, csc x, cot x |
| eˣ, ln x, aˣ, logₐ x | Derivatives requiring product/quotient/chain rules |
You must memorise the derivatives of sec x, csc x, and cot x, as they are NOT in the IB formula booklet.
你必须记忆 sec x、csc x 和 cot x 的导数，因为它们在IB公式手册中不存在。
11. The Derivative Table in AI Contexts | AI课程中的导数表应用
In Applications & Interpretation, differentiation appears primarily in the context of rates of change, optimisation in real-world models, and understanding the behaviour of functions.
在应用与解释（AI）课程中，微分主要出现在变化率、实际模型中的优化问题以及理解函数行为的情境中。
- Example: he : The cost function for producing x units is C(x) = 500 + 20x + 0.1x². The marginal cost is C'(x) = 20 + 0.2x.
- 例子： 生产 x 件产品的成本函数为 C(x) = 500 + 20x + 0.1x²。边际成本为 C'(x) = 20 + 0.2x。
- Interpretation: 20 + 0.2(50) = 30, meaning the cost of producing one additional unit is approximately 30.
- 解读： 当 x = 50 时，边际成本为 20 + 0.2(50) = 30，意味着多生产一件产品的成本约为30。
AI students do not need to memorise as many derivatives as AA students, but the core rules (power, exponential, trigonometric) are still essential.
AI学生比AA学生需要记忆的导数要少一些，但核心法则（幂、指数、三角）仍然必不可少。
12. A Comprehensive Practice Set | 综合练习集
Test your mastery of the basic derivative table with the following questions. Attempt each one before revealing the answer.
用以下问题测试你对基本导数表的掌握程度。先尝试作答，再查看答案。
- Q1: Differentiate f(x) = 3x⁵ − 2x³ + x − 7
- 问1： 求导 f(x) = 3x⁵ − 2x³ + x − 7
- Q2: Differentiate g(t) = 4sin(t) + 5cos(t)
- 问2： 求导 g(t) = 4sin(t) + 5cos(t)
- Q3: Differentiate h(x) = x²·eˣ
- 问3： 求导 h(x) = x²·eˣ
- Q4: Differentiate y = ln(3x² + 1)
- 问4： 求导 y = ln(3x² + 1)
Answers: 1) f'(x) = 15x⁴ − 6x² + 1 2) g'(t) = 4cos(t) − 5sin(t) 3) h'(x) = x·eˣ(x + 2) 4) y' = 6x/(3x² + 1)
If you got all four correct, you have mastered the essentials. If not, revisit the relevant sections above and practise regularly.
如果你四题全部正确，说明你已经掌握了核心内容。如果没有，请重读上文相关章节并定期练习。
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