Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的求导

📚 Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的求导

Exponential and logarithmic functions are among the most fundamental functions in mathematics, appearing throughout A-Level Mathematics and Further Mathematics syllabi. This article provides a systematic, examination-focused guide to differentiating these functions, from the natural exponential and natural logarithm to general bases, implicit differentiation, and logarithmic differentiation.

指数函数与对数函数是数学中最基本的函数类型之一，贯穿于A-Level数学与进阶数学的全部考纲。本文旨在提供一篇系统化、紧扣考点的求导指南，内容涵盖自然指数函数、自然对数函数、一般底数函数、隐函数求导法以及对数微分法等内容。

1. The Natural Exponential Function eˣ | 自然指数函数 eˣ

The function eˣ, where e ≈ 2.71828 is Euler’s number, is the unique function whose derivative is itself. This remarkable property makes e the natural base for calculus.

函数 eˣ（其中 e ≈ 2.71828 为欧拉数）是唯一一个导函数等于自身的函数。这一非凡性质使 e 成为微积分中的自然底数。

d/dx (eˣ) = eˣ

This means the gradient of the curve y = eˣ at any point x is exactly equal to the y-coordinate of that point. For example, at x = 0, the gradient is e⁰ = 1; at x = 2, the gradient is e² ≈ 7.39.

这意味着曲线 y = eˣ 在任意点 x 处的斜率恰好等于该点的 y 坐标。例如，在 x = 0 处，斜率为 e⁰ = 1；在 x = 2 处，斜率为 e² ≈ 7.39。

2. The Chain Rule for e^u | e^u 的链式法则

When the exponent is itself a function of x, we apply the chain rule. If y = e^u where u = f(x), then:

当指数本身是 x 的函数时，我们需要运用链式法则。若 y = e^u，其中 u = f(x)，则：

dy/dx = e^u · du/dx

For example, if y = e^{3x}, then u = 3x, so dy/dx = e^{3x} · 3 = 3e^{3x}. If y = e^{x²}, then dy/dx = 2x·e^{x²}.

例如，若 y = e^{3x}，则 u = 3x，因此 dy/dx = e^{3x} · 3 = 3e^{3x}。若 y = e^{x²}，则 dy/dx = 2x·e^{x²}。

General form: d/dx (e^{f(x)}) = f'(x) · e^{f(x)}

This rule can be remembered as ‘the derivative of the exponent, multiplied by the original function’. It is essential to identify the inner function clearly before differentiating.

这条法则可以记忆为“指数部分的导数乘以原函数”。在求导之前，必须先明确识别内层函数的结构。

3. The Natural Logarithm ln x | 自然对数 ln x

The natural logarithm ln x is defined as the inverse function of eˣ. Since e^ln x = x for all x > 0, we can use implicit differentiation to find its derivative.

自然对数 ln x 定义为 eˣ 的反函数。由于对所有 x > 0 都有 e^ln x = x，我们可以利用隐函数求导法来推导它的导数。

Let y = ln x. Then eʸ = x. Differentiating both sides with respect to x:

设 y = ln x，则 eʸ = x。两边对 x 求导：

eʸ · dy/dx = 1 → dy/dx = 1/eʸ = 1/x

d/dx (ln x) = 1/x (for x > 0)

Equivalently, the slope of y = ln x at any point x is the reciprocal of x. The derivative is always positive, confirming ln x is a strictly increasing function on its domain.

换言之，y = ln x 在任意一点 x 处的斜率等于 x 的倒数。导数始终为正，这证实了 ln x 在其定义域上是严格递增函数。

4. The Chain Rule for ln u | ln u 的链式法则

If u = f(x) > 0 and y = ln u, then by the chain rule:

若 u = f(x) > 0 且 y = ln u，根据链式法则：

dy/dx = (1/u) · du/dx = u’/u

For example, if y = ln(2x + 1), then dy/dx = 2/(2x + 1). If y = ln(sin x), then dy/dx = cos x / sin x = cot x.

例如，若 y = ln(2x + 1)，则 dy/dx = 2/(2x + 1)。若 y = ln(sin x)，则 dy/dx = cos x / sin x = cot x。

A particularly useful application is the derivative of ln|x|. For x > 0, d/dx (ln|x|) = 1/x; for x < 0, we also obtain 1/x. Hence the formula d/dx (ln|x|) = 1/x holds for all x ≠ 0, which is used to integrate 1/x over negative intervals.

一个特别有用的应用是 ln|x| 的导数。当 x > 0 时，d/dx (ln|x|) = 1/x；当 x < 0 时，同样得到 1/x。因此公式 d/dx (ln|x|) = 1/x 对所有 x ≠ 0 都成立，这使得我们可以在负数区间上对 1/x 进行积分。

5. General Exponential Functions aˣ | 一般指数函数 aˣ

For any positive base a ≠ 1, the function aˣ can be written as e^{x·ln a}. This allows us to derive its derivative:

对于任意正底数 a ≠ 1，函数 aˣ 都可以写成 e^{x·ln a} 的形式。这使得我们能够推导出它的导数：

d/dx (aˣ) = aˣ · ln a

To prove this, note that aˣ = e^{x·ln a}. Applying the chain rule with u = x·ln a gives d/dx (aˣ) = e^{x·ln a} · ln a = aˣ · ln a.

证明如下：由于 aˣ = e^{x·ln a}，运用链式法则，令 u = x·ln a，则 d/dx (aˣ) = e^{x·ln a} · ln a = aˣ · ln a。

When a = e, this reduces to d/dx (eˣ) = eˣ · ln e = eˣ. When a = 2, d/dx (2ˣ) = 2ˣ·ln 2, which appears frequently in Cambridge Edexcel-style examination questions involving exponential growth models.

当 a = e 时，公式简化为 d/dx (eˣ) = eˣ · ln e = eˣ。当 a = 2 时，d/dx (2ˣ) = 2ˣ·ln 2，这个结果常见于涉及指数增长模型的考试题中。

6. General Logarithmic Functions logₐx | 一般对数函数 logₐx

For a general base a, the logarithm logₐx can be expressed in terms of the natural logarithm using the change of base formula:

对于一般底数 a，对数 logₐx 可以通过换底公式用自然对数来表示：

logₐx = ln x / ln a

Since ln a is a constant, differentiating both sides gives:

由于 ln a 是常数，两边求导得到：

d/dx (logₐx) = 1/(x·ln a)

For example, d/dx (log₁₀x) = 1/(x·ln 10). Note that log₁₀x is often written as lg x in the Chinese curriculum, and this derivative identity is a frequent exam favourite.

例如，d/dx (log₁₀x) = 1/(x·ln 10)。注意 log₁₀x 在中国的课程中常写作 lg x，这个导数公式也是考试中的常见考点。

7. Implicit Differentiation with Exponentials and Logarithms | 含指数与对数的隐函数求导

When y cannot be expressed explicitly as a function of x, we use implicit differentiation. This approach is especially powerful when dealing with exponentials and logarithms.

当 y 无法显式表示为 x 的函数时，我们使用隐函数求导法。这种方法在处理指数和对数问题时尤为强大。

For the function y = e^{x+y}, differentiating both sides with respect to x yields:

对于函数 y = e^{x+y}，两边对 x 求导得到：

dy/dx = e^{x+y} · (1 + dy/dx)

Since e^{x+y} = y, we can substitute to simplify: dy/dx = y(1 + dy/dx), which rearranges to dy/dx = y/(1 − y).

由于 e^{x+y} = y，我们可以代入化简：dy/dx = y(1 + dy/dx)，整理得 dy/dx = y/(1 − y)。

This technique is frequently tested in A-Level papers, particularly in Part (b) of long-form questions where students must find the gradient at a given point without solving for y explicitly.

这种技巧在A-Level试卷中频繁出现，特别是在大题的第 (b) 小问中，要求学生不求 y 的显式表达式、直接求指定点处的斜率。

8. Logarithmic Differentiation | 对数微分法

Logarithmic differentiation is a powerful technique for differentiating functions that involve products, quotients, powers, or exponents. The steps are as follows:

对数微分法是一种用于求导涉及乘积、商、幂或指数结构函数的强大技巧。其步骤如下：

Take the natural logarithm of both sides of y = f(x).
对等式两边同时取自然对数：ln y = ln f(x)。
Simplify using logarithm laws: ln(ab) = ln a + ln b, ln(aᵇ) = b·ln a.
利用对数法则化简：ln(ab) = ln a + ln b，ln(aᵇ) = b·ln a。
Differentiate both sides implicitly with respect to x.
两边对 x 隐函数求导。
Multiply both sides by y to solve for dy/dx.
两边同时乘以 y，解出 dy/dx。

For example, differentiate y = xˣ. Taking logs: ln y = x·ln x. Differentiating: (1/y)·dy/dx = ln x + 1. Hence dy/dx = y(ln x + 1) = xˣ(ln x + 1).

例如，求 y = xˣ 的导数。取对数：ln y = x·ln x。求导：(1/y)·dy/dx = ln x + 1。因此 dy/dx = y(ln x + 1) = xˣ(ln x + 1)。

d/dx (xˣ) = xˣ(ln x + 1)

This method is also ideal for differentiating functions such as y = x^sin x, y = (x²+1)/(x−1)³, or any expression where both part and index contain variables.

此方法同样适用于求导如 y = x^sin x、y = (x²+1)/(x−1)³ 等底数和指数同时含变量的函数表达式。

9. Key Derivative Table | 核心导数对照表

Function \| 函数	Derivative \| 导数
eˣ	eˣ
e^{f(x)}	f'(x) · e^{f(x)}
aˣ	aˣ · ln a
ln x	1/x
ln\|x\|	1/x
ln f(x)	f'(x)/f(x)
logₐx	1/(x·ln a)
xˣ	xˣ(ln x + 1)

This table summarises the essential derivative formulae. Students should memorise the first, third, fourth, and sixth rows as a priority — they are the most frequently examined.

此表总结了所有必备的导数公式。学生应当优先记住第 1、3、4、6 行，因为它们是考试中出现频率最高的公式。

10. Applications: Tangents, Normals and Rates | 应用：切线、法线与变化率

The derivative of an exponential or logarithmic function at a point gives the gradient of the tangent at that point. This leads directly to the equation of the tangent or normal.

指数或对数函数在某一点的导数给出了该点切线的斜率，进一步可直接写出切线或法线方程。

For example, find the equation of the tangent to y = e^{2x} at x = 0. Since dy/dx = 2e^{2x}, at x = 0 we have dy/dx = 2 and y = 1. The tangent is y − 1 = 2(x − 0), i.e. y = 2x + 1.

例如，求曲线 y = e^{2x} 在 x = 0 处的切线方程。由于 dy/dx = 2e^{2x}，在 x = 0 处 dy/dx = 2 且 y = 1。切线方程为 y − 1 = 2(x − 0)，即 y = 2x + 1。

Exponential and logarithmic derivatives are also central to rates of change problems. For example, in modelling growth of a population N(t) = N₀e^{kt}, the growth rate is dN/dt = kN₀e^{kt} = kN(t), showing a constant proportionality k.

指数和对数导数同样在变化率问题中扮演核心角色。例如，在群体增长模型 N(t) = N₀e^{kt} 中，增长率为 dN/dt = kN₀e^{kt} = kN(t)，表明增长速率与当前群体大小成正比，比例常数为 k。

11. Common Mistakes and Exam Tips | 常见错误与备考提示

Students frequently make the following mistakes when differentiating exponential and logarithmic functions:

学生在对指数函数和对数函数求导时，常犯以下错误：

Forgetting the chain rule: writing d/dx(e^{2x}) = e^{2x} instead of 2e^{2x}. Remember to multiply by the derivative of the exponent.
忘记链式法则：将 d/dx(e^{2x}) 误写为 e^{2x} 而漏乘 2。务必乘以指数部分的导数。
Confusing aˣ with ax: the derivative of aˣ requires multiplying by ln a, while the derivative of ax is simply a.
混淆 aˣ 与 ax：aˣ 的导数需要乘以 ln a，而 ax 的导数仅为 a。
Writing d/dx(ln(3x)) = 1/(3x): the correct answer is 3/(3x) = 1/x. The factor from the chain rule cancels with the denominator.
误写 d/dx(ln(3x)) = 1/(3x)：正确答案为 3/(3x) = 1/x。链式法则产生的因子与分母约去。
Forgetting to take logs before differentiating xˣ: the power rule only applies when the exponent is a constant.
对 xˣ 求导时忘记先取对数：幂函数求导法则仅在指数为常数时成立。
Forgetting that ln x is only defined for x > 0.
忘记 ln x 仅在 x > 0 时有定义。

Exam strategies: (1) Always state the rule you are using, as examiners award method marks. (2) Simplify algebraic expressions before differentiating where possible. (3) Check your answer by verifying units and considering sign behaviour.

考试策略：(1) 始终指明你所用的法则，阅卷人会根据方法步骤给分。(2) 在求导前尽可能先化简代数表达式。(3) 通过单位和符号行为检查答案的合理性。

12. Summary | 总结

The key results of this article are the fundamental differential relationships between exponentials and logarithms: d/dx(eˣ) = eˣ, d/dx(ln x) = 1/x, and their generalised forms via the chain rule and implicit differentiation. Understanding these rules allows students to differentiate any combination of exponential and logarithmic functions confidently.

本文的核心结果是指数函数与对数函数之间的基本微分关系：d/dx(eˣ) = eˣ、d/dx(ln x) = 1/x，以及通过链式法则和隐函数求导得到的推广形式。理解这些法则，能够帮助学生自信地求导任意涉及指数函数和对数函数的表达式。

Mastery of these techniques is not only essential for A-Level examinations but also forms the foundation for future study of differential equations, series expansions, and mathematical modelling in economics and the natural sciences.

掌握这些技巧不仅对A-Level考试至关重要，更将为后续学习微分方程、级数展开以及经济学和自然科学中的数学建模奠定基础。

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Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的求导