📚 Mastering Differentiation of Exponential and Logarithmic Functions | 掌握指数与对数函数的微分
Differentiation of exponential and logarithmic functions is a cornerstone of A-Level Mathematics. In Edexcel's Pure Mathematics specification, you are expected to differentiate expressions involving eˣ, ln x, aˣ, and logₐ x fluently, and to combine these skills with the chain, product, and quotient rules. This article develops each rule step by step, provides worked examples, and highlights common pitfalls to help you secure top marks.
指数函数与对数函数的微分是 A-Level 数学的核心内容。在 Edexcel 纯数学考纲中，不仅要熟练对 eˣ、ln x、aˣ 和 logₐ x 求导，还必须能灵活结合链式法则、乘积法则和商法则处理复合函数。本文将逐条建立求导规则、给出范例并指出常见错误，帮助你稳拿高分。
1. The Exponential and Logarithmic Functions We Need | 我们需要掌握的指数与对数函数
The natural exponential function f(x) = eˣ elf. The natural logarithm, ln x, is its inverse. We also work with general exponentials aˣ e relationship between these functions is essential before applying differentiation rules.
自然指数函数 f(x) = eˣ 的独特性质是它的导数等于它本身。自然对数 ln x 是它的反函数。我们还会遇到一般的指数形式 aˣ（a > 0, a ≠ 1）及其反函数 logₐ x。在应用求导规则之前，认清这些函数之间的关系至关重要。
2. Derivative of eˣ – The Simplest Case | eˣ 的导数——最简单的情形
The derivative of eˣ is eˣ. This means the gradient at any point equals the function value at that point. Symbolically:
d/dx(eˣ) = eˣ
No coefficient changes; the derivative remains exactly eˣ. This result underpins all further work with exponential functions.
eˣ 的导数为 eˣ。也就是说，任何点的斜率都等于该点的函数值。这是指数函数微分所有后续推导的基础。
3. Derivative of aˣ – General Exponential | aˣ 的导数——一般指数函数
When the base is a positive constant a (a ≠ 1), the derivative includes an extra factor ln a:
d/dx(aˣ) = aˣ ln a
This can be derived from eˣ by rewriting aˣ = ex ln a and applying the chain rule. Always remember the ln a multiplier – it is one of the most frequently omitted pieces in exams.
当底数为正常数 a 且 a ≠ 1 时，导数中包含额外的因子 ln a。可通过将 aˣ 写作 ex ln a 并结合链式法则推导得到。考试中遗忘 ln a 乘数是最常见的失分点之一。
4. Derivative of ln x – The Natural Logarithm | ln x 的导数——自然对数
The derivative of ln x is 1/x, defined for x > 0:
d/dx(ln x) = 1/x
This result comes from differentiating the inverse relation eln x in restriction: ln x is only differentiable for positive x, and the derivative 1/x is always positive where defined.
ln x 的导数是 1/x（要求 x > 0）。该结果来自对反函数关系 eln x = x 进行隐函数微分。千万注意定义域限制：只有正实数范围内 ln x 可导，且其导数始终为正。
5. Derivative of logₐ x – General Logarithm | logₐ x 的导数——一般对数函数
For a logarithm with base a, the derivative introduces a factor 1/(ln a) alongside the familiar 1/x:
d/dx(logₐ x) = 1/(x ln a)
ng the change-of-base formula (logₐ x = ln x / ln a) makes the derivation straightforward.
以 a 为底的对数求导时，除了 1/x 外还需要分母上乘一个 ln a 的倒数。Edexcel 常考类似 log₂ x 的求导，利用换底公式 logₐ x = ln x / ln a 可以轻松推导。
6. Chain Rule with ef(x) | ef(x) 的链式法则
When the exponent is a function of x, multiply by the derivative of that exponent:
d/dx(ef(x)) = f'(x) · ef(x)
For example, d/dx(e3x) = 3e3x; for ex² the derivative is 2x·ex². The key is to differentiate the power first, then keep the original exponential unchanged.
当指数是 x 的函数时，将指数函数的导数乘以该指数的导数即可。例如 d/dx(e3x) = 3e3x；对于 ex²，导数为 2x·ex²。要点是：先对指数求导，保留原指数函数不变。
7. Chain Rule with ln(f(x)) | ln(f(x)) 的链式法则
Differentiating a logarithm of a function gives a ‘derivative over function’ pattern:
d/dx(ln(f(x))) = f'(x) / f(x)
(5x) = 1/x, which is consistent because ln(5x) = ln 5 + ln x, and the constant disappears.
对对数复合函数求导会得到“导数除以函数本身”的形式。这一表达式在后续积分中经常出现。例如 d/dx(ln(5x)) = 5/(5x) = 1/x，与 ln(5x) = ln 5 + ln x 的结果一致，因为常数导数为零。
8. Product and Quotient Rules with Exponentials and Logs | 乘积法则与商法则结合指数、对数
ions, requiring both chain and product/quotient rules. For example, differentiate y = x²·eˣ. By the product rule:
dy/dx = 2x·eˣ + x²·eˣ = eˣ(2x + x²)
Similarly, for y = ln x / x, apply the quotient rule:
dy/dx = ((1/x)·x - ln x·1) / x² = (1 - ln x) / x²
Always identify which rule applies first, then work systematically layer by layer.
很多考题将指数或对数与代数函数混合，要求同时使用链式法则和乘积/商法则。例如 y = x²·eˣ，用乘积法则得 dy/dx = 2x·eˣ + x²·eˣ。对于 y = ln x / x，用商法则得到 dy/dx = (1 - ln x) / x²。解题时要先确定规则顺序，分层计算。
9. Differentiating More Complex Combinations | 复合函数的多层微分
Consider y = ln(eˣ + e⁻ˣ). Here the outer function is ln u, the inner is eˣ + e⁻ˣ. Differentiation proceeds stepwise:
- Outer derivative: 1/(eˣ + e⁻ˣ)
- Inner derivative: eˣ - e⁻ˣ
- Overall: dy/dx = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ)
Always check whether the result can be simplified; here it cannot easily be reduced, but recognising it as tanh x may be useful in further study.
考虑 y = ln(eˣ + e⁻ˣ)。外层函数为 ln u，内层为 eˣ + e⁻ˣ。分层求导：外层导数 1/(eˣ + e⁻ˣ)，内层导数 eˣ - e⁻ˣ，组合得 dy/dx = (eˣ - e⁻ˣ) / (eˣ + e⁻ˣ)。可以检验可否化简；此处结果可识为双曲正切 tanh x，为后续学习提供便利。
10. Applications: Tangents and Normals | 应用：切线与法线
nts and normals to exponential/logarithmic curves is a staple exam task. For a curve y = e2x at x = 0:
- Point: (0, 1)
- Gradient: dy/dx = 2e2x ⇒ at x=0 gradient = 2
- Tangent: y - 1 = 2(x - 0) ⇒ y = 2x + 1
- Normal: gradient = -½ ⇒ y - 1 = -½(x - 0) ⇒ y = -½x + 1
Always find the y-coordinate first, then the gradient at that point. Many marks are lost by using the derivative expression without evaluating it.
求指数或对数曲线的切线和法线方程是常见考题。以 y = e2x 在 x=0 处为例：先确定点 (0,1)，导数为 2e2x，代入得斜率 2；切线方程 y = 2x + 1；法线斜率为 -½，方程 y = -½x + 1。务必先求 y 坐标和该点斜率，许多考生因未代入数值而失分。
11. Common Mistakes and How to Avoid Them | 常见错误与避免方法
Top mistakes include: forgetting the ln a factor in d/dx(aˣ); omitting the '× inner derivative' step with ef(x); mishandling domains of ln(f(x)) (f(x) must be >0); confusing product and chain rules; and attempting to differentiate eˣ as if it were a power function. A disciplined approach with clear working prevents these.
高频错误包括：遗忘 d/dx(aˣ) 中的 ln a 因子；省略 ef(x) 中“乘以内部导数”的步骤；错误处理 ln(f(x)) 的定义域（f(x) 必须大于 0）；混淆乘积法则与链式法则；以及将 eˣ 错误地当作幂函数求导。保持步骤分明、书写规范即可有效避免这些失误。
12. Exam-Style Mixed Question | 考试风格混合题示例
x³·ln(2x+1) with respect to x. Identify it as a product where u = x³, v = ln(2x+1). Then u' = 3x², v' = 2/(2x+1) by the chain rule. Therefore:
dy/dx = 3x²·ln(2x+1) + x³·(2/(2x+1))
State the domain explicitly: 2x+1 > 0 ⇒ x > -½. Edexcel mark schemes often reward domain statements alongside the derivative.
对 y = x³·ln(2x+1) 求导。识别为乘积：设 u = x³, v = ln(2x+1)。则 u' = 3x²，使用链式法则得 v' = 2/(2x+1)。组合得 dy/dx = 3x²·ln(2x+1) + 2x³/(2x+1)。明确指出定义域：2x+1 > 0 ⇒ x > -½。Edexcel 评分标准常对注明定义域给予加分。
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