📚 Derivative of Logarithmic Functions | 对数函数的导数
Logarithmic functions appear constantly in mathematics, physics, economics, and the natural sciences. Understanding how to differentiate them is therefore a fundamental skill for any IB Mathematics student. In this article, we will explore the derivative of both the natural logarithm and logarithms of any base, together with the chain rule, logarithmic differentiation, and important applications.
对数函数在数学、物理、经济学和自然科学中随处可见。因此，掌握对数函数的求导方法是每一位 IB 数学学生必备的基础技能。本文将系统讲解自然对数以及任意底数对数的导数，并结合链式法则、对数求导法及其重要应用进行深入探讨。
1. The Natural Logarithm | 自然对数
The natural logarithm, denoted ln x (careful — we write ln x but never use LaTeX; here we simply write ln x), is the inverse of the exponential function ex. Its domain is x > 0, and its range is all real numbers.
自然对数记作 ln x，是指数函数 ex 的反函数。其定义域为 x > 0，值域为全体实数。
d/dx [ln x] = 1/x, for x > 0
This formula is the cornerstone of all logarithmic differentiation. It is simple to memorise but subtle to apply correctly, especially when the argument of the logarithm is a function rather than just x.
这一公式是对数求导的基石。它易于记忆，但在自变量不是单纯的 x 而是一个函数时，正确运用却需要多加注意。
2. Intuition Behind the Formula | 公式背后的直觉
Why is the derivative of ln x equal to 1/x? One intuitive way to see this is to observe that the slope of the tangent to y = ln x becomes less steep as x increases, which matches the decreasing positive values of 1/x.
为什么 ln x 的导数等于 1/x？一种直观的理解方式是：随着 x 增大，曲线 y = ln x 的切线斜率逐渐变缓，这与 1/x 递减的正值相吻合。
Alternatively, since y = ln x is the inverse of x = ey, we can use the inverse function theorem:
另一种方式是利用反函数定理。因为 y = ln x 是 x = ey 的反函数：
dx/dy = ey = x, therefore dy/dx = 1/x
This derivation is elegant and reinforces the relationship between exponential and logarithmic functions.
这一推导十分简洁，也再次印证了指数函数与对数函数之间的密切联系。
3. Derivative of ln |u| | ln |u| 的导数
A very useful extension is the derivative of the natural logarithm of an absolute value:
一个非常有用的推广形式是自然对数绝对值的导数：
d/dx [ln |x|] = 1/x, for x ≠ 0
Notice that the absolute value extends the domain to all non-zero real numbers, while the derivative formula remains the same. This is particularly helpful when differentiating expressions involving even powers or other quantities that might be negative.
请注意，取绝对值后将定义域扩展到了所有非零实数，而导数公式保持不变。当处理偶次幂或其他可能为负的量时，这一性质尤其有用。
4. General Logarithmic Base | 一般底数的对数
What if the base is not e? For a general base a > 0, a ≠ 1, we use the change of base formula:
如果底数不是 e 该怎么办？对于一般底数 a > 0 且 a ≠ 1，我们使用换底公式：
logₐ x = ln x / ln a
Since ln a is a constant, differentiating both sides gives:
由于 ln a 是常数，对等式两边求导可得：
d/dx [logₐ x] = 1/(x ln a)
This result is essential when dealing with logarithms in bases other than e, such as log₁₀ or log₂.
当处理 e 以外的底数（如 log₁₀ 或 log₂）时，这个结果至关重要。
5. Applying the Chain Rule | 链式法则的应用
When the argument of a logarithm is a function u(x) > 0, the chain rule takes the form:
当对数的自变量是一个函数 u(x) > 0 时，链式法则表现为：
d/dx [ln u(x)] = u'(x) / u(x)
In words, the derivative is the derivative of the inside function divided by the inside function itself. This pattern appears over and over again in IB exams.
用语言来描述就是：导数为内部函数的导数除以内部函数本身。这个模式在 IB 考试中反复出现。
Let us work through an example. If y = ln (3x² + 2), then:
我们来具体算一个例子。若 y = ln (3x² + 2)，那么：
dy/dx = 6x / (3x² + 2)
Notice that the numerator 6x is exactly the derivative of 3x² + 2. This method avoids the need to use the definition of the derivative each time.
注意分子 6x 恰好是 3x² + 2 的导数。这种方法让我们无须每次都从导数定义出发。
6. Logarithmic Differentiation | 对数求导法
Logarithmic differentiation is a powerful technique that simplifies derivatives of products, quotients, and powers. For a function y = f(x), we first take the natural logarithm of both sides and then differentiate implicitly.
对数求导法是一种强大的技巧，可以大大简化乘积、商和幂函数的求导。对于函数 y = f(x)，我们首先对等式两边取自然对数，然后进行隐函数求导。
Example: find the derivative of y = xx for x > 0. First take ln of both sides:
例如：求 y = xx（x > 0）的导数。首先对两边取对数：
ln y = ln (xx) = x ln x
Differentiating both sides implicitly:
对两边进行隐函数求导：
(1/y) dy/dx = ln x + 1, therefore dy/dx = xx (ln x + 1)
Notice how the product rule appears naturally through the term x ln x. This technique is particularly valuable for functions where the variable appears both in the base and the exponent.
注意乘积法则如何通过 x ln x 这一项自然出现。当变量同时出现在底数和指数中时，这种方法尤其有效。
7. Summary of Key Formulas | 关键公式汇总
The following table summarises the essential derivative formulas for logarithmic functions. Keep this table handy when revising for the IB exam.
下表汇总了对数函数求导的核心公式。复习 IB 考试时，请将此表放在手边。
| Function | Derivative |
| ln x | 1/x |
| ln |x| | 1/x |
| ln u(x) | u'(x) / u(x) |
| logₐ x | 1/(x ln a) |
| logₐ u(x) | u'(x) / (u(x) ln a) |
8. Common Mistakes to Avoid | 需要避免的常见错误
Students often make a few predictable mistakes when differentiating logarithms. Being aware of these pitfalls will help you avoid losing marks in the exam.
学生在求对数导数时经常会犯一些可预见的错误。意识到这些陷阱将帮助你在考试中避免失分。
Mistake 1: Forgetting that the derivative of ln u is u'/u, not 1/u. Always apply the chain rule.
错误 1：忘记 ln u 的导数是 u'/u，而不是 1/u。务必使用链式法则。
Mistake 2: Using ln (a + b) = ln a + ln b. This is false. Addition inside a logarithm does not split in this way.
错误 2：误认为 ln (a + b) = ln a + ln b。这是错误的。对数内部的加法不能这样拆分。
Mistake 3: Forgetting the domain restriction x > 0 for ln x. When writing the final derivative, always mention the domain.
错误 3：忘记 ln x 的定义域为 x > 0。在写出最终导数时，务必注明定义域。
Mistake 4: Confusing the derivative of logₐ x (which has 1/ln a) with the derivative of ax (which has ln a). They are different patterns!
错误 4：混淆 logₐ x 的导数（含有 1/ln a）与 ax 的导数（含有 ln a）。两者结构完全不同！
9. Worked IB-Style Examples | IB 风格例题精讲
Let us now apply the theory to three typical IB examination problems.
下面我们将理论应用于三道典型的 IB 考试问题。
Example 1: Differentiate y = log₃ (2x + 1). Using the formula for general base and the chain rule:
例 1：求 y = log₃ (2x + 1) 的导数。利用一般底数公式及链式法则：
dy/dx = 2 / [(2x + 1) ln 3]
Here the numerator 2 is the derivative of 2x + 1, and the denominator contains (2x + 1) ln 3.
其中分子 2 是 2x + 1 的导数，分母为 (2x + 1) ln 3。
Example 2: Find the derivative of y = ln √x² + 1. First simplify using logarithm laws:
例 2：求 y = ln √x² + 1 的导数。先用对数性质化简：
y = ½ ln (x² + 1)
Then differentiate:
然后求导：
dy/dx = ½ × [2x / (x² + 1)] = x / (x² + 1)
Simplifying the original expression first often makes the differentiation far easier.
先化简原表达式往往能让求导过程轻松许多。
Example 3: Use logarithmic differentiation to find dy/dx for y = (sin x)cos x. Taking ln of both sides:
例 3：利用对数求导法求 y = (sin x)cos x 的导数。对两边取对数：
ln y = cos x · ln (sin x)
Differentiating implicitly:
隐式求导：
(1/y) dy/dx = –sin x · ln (sin x) + cos x · (cos x / sin x)
dy/dx = (sin x)cos x [ cos² x / sin x – sin x ln (sin x) ]
This example demonstrates how logarithmic differentiation handles complicated exponentiated functions elegantly.
此例展示了对数求导法如何优雅地处理复杂的指数型函数。
10. Relationship with Exponential Derivatives | 与指数函数导数的联系
The derivative of ln x and the derivative of ex are intimately connected through the general rule for inverse functions. If y = ex, then x = ln y, and:
ln x 的导数与 ex 的导数通过反函数法则紧密相连。若 y = ex，则 x = ln y，于是：
dy/dx = ex, and indeed (1/y) dy/dx = 1 implies dy/dx = y = ex
In IB, it is common to see questions that ask you to differentiate functions combining exponentials and logarithms, so keep both formulas fresh in your mind.
在 IB 考试中，常见题目会要求你求指数与对数组合函数的导数，因此务必牢记两套公式。
11. Applications in Growth and Decay | 在增长与衰减中的应用
Logarithmic derivatives appear naturally in the study of exponential growth and decay. Consider a population P(t) = P₀ ekt. Taking logs gives ln P = ln P₀ + kt. The derivative d/dt [ln P] = P'/P represents the relative rate of growth.
对数导数在指数增长与衰减的研究中自然出现。考虑种群数量 P(t) = P₀ ekt。取对数得 ln P = ln P₀ + kt。导数 d/dt [ln P] = P'/P 表示相对增长率。
This concept — the derivative divided by the function itself — is called the logarithmic derivative and appears in economics (elasticity), biology (population growth), and physics (radioactive decay).
这种“导数除以函数本身”的概念被称为对数导数，在经济学（弹性）、生物学（种群增长）和物理学（放射性衰变）中都有重要应用。
12. Final Tip for the Exam | 考试终极提示
Before you close your notebook, remember these three steps when you encounter a logarithmic derivative:
在合上笔记本之前，请记住遇到对数求导时的三个步骤：
Identify the base. If it is e, use 1/u × u'. If it is another base a, multiply by 1/ln a.
识别底数。如果是 e，则用 1/u × u'；如果是其他底数 a，则乘以 1/ln a。
Simplify with logarithm laws wherever possible, such as turning powers into coefficients.
尽可能利用对数性质化简，例如将幂变成系数。
Always state the domain restriction (u > 0) when writing your final answer.
写出最终答案时，务必注明定义域限制（u > 0）。
With consistent practice, logarithmic differentiation will become second nature. Good luck with your IB Mathematics revision!
通过持续练习，对数求导必将变得得心应手。祝你 IB 数学复习顺利！
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图