📚 Differentiation of Exponential and Logarithmic Functions | 指数函数与对数函数的求导
In Edexcel A-Level Mathematics, differentiating exponential and logarithmic functions is a fundamental skill, especially in Pure Mathematics 2 and 3. Understanding the derivatives of eˣ, ln x, aˣ, and applying the chain, product, and quotient rules enables students to solve a wide variety of problems, from growth and decay models to complex function analysis. This article provides a comprehensive overview of the key rules, worked examples, and exam tips.
在 Edexcel A-Level 数学中，指数函数与对数函数的求导是基础技能，尤其体现在 Pure Mathematics 2 和 3 中。掌握 eˣ、ln x、aˣ 的导数，并灵活运用链式法则、乘法法则和除法法则，可以帮助学生解决各种问题，从增长与衰减模型到复杂函数分析。本文全面梳理关键法则、典型例题与考试技巧。
1. Derivative of eˣ | eˣ 的导数
The exponential function f(x) = eˣ has the unique property that it is its own derivative. That is, d/dx (eˣ) = eˣ nge equal to its value at every point. When e appears with a constant multiple, such as eᵏˣ, the chain rule is used.
指数函数 f(x) = eˣ 具有一个独特性质：其导数等于本身，即 d/dx(eˣ) = eˣ。这是因为自然指数函数在每一点的变化率等于其函数值。当 e 带有常数倍，如 eᵏˣ 时，需要使用链式法则。
For example, differentiate e³ˣ. Let u = 3x, then dy/du = eᵘ, du/dx = 3, so dy/dx = 3e³ˣ. In general, d/dx (eᵏˣ) = k eᵏˣ.
例如，求 e³ˣ 的导数。设 u = 3x，则 dy/du = eᵘ，du/dx = 3，因此 dy/dx = 3e³ˣ。一般地，d/dx(eᵏˣ) = k eᵏˣ。
2. Derivative of ln x | ln x 的导数
0. When the argument is a function of x, use the chain rule: d/dx (ln u) = (1/u) × du/dx.
自然对数函数 f(x) = ln x 的导数为 d/dx(ln x) = 1/x，其中 x > 0。当真数为 x 的函数时，应用链式法则：d/dx(ln u) = (1/u) × du/dx。
(2x+1)) × 2 = 2/(2x+1). This rule is essential for integrating rational functions as well.
示例：求 ln(2x+1) 的导数。设 u = 2x+1，则导数为 (1/(2x+1)) × 2 = 2/(2x+1)。这一法则对有理函数的积分也至关重要。
3. Derivative of aˣ (Exponential with Base a) | aˣ 的导数（以 a 为底的指数函数）
For a general exponential function with base a > 0, a ≠ 1, the derivative is d/dx(aˣ) = aˣ ln a. This can be derived by writing aˣ = eˣ ˡⁿ ᵃ and differentiating.
对于底数 a > 0 且 a ≠ 1 的一般指数函数，其导数为 d/dx(aˣ) = aˣ ln a。这一结果可通过将 aˣ 写为 eˣ ln a 再求导得出。
Example: f(x) = 2ˣ, then f'(x) = 2ˣ ln 2. Using the chain rule, d/dx(aᵘ) = aᵘ ln a × du/dx.
例如：f(x) = 2ˣ，则 f'(x) = 2ˣ ln 2。结合链式法则，d/dx(aᵘ) = aᵘ ln a × du/dx。
4. Derivative of logₐ x | logₐ x 的导数
converting to natural log: logₐ x = ln x / ln a, so its derivative is (1/(ln a)) × (1/x).
以 a 为底的对数函数导数为 d/dx(logₐ x) = 1/(x ln a)。这可通过换底公式 logₐ x = ln x / ln a 求导得出，即 (1/ln a) × (1/x)。
1/(x ln 10). In exam questions, students may be asked to differentiate log₂ x or similar.
例如 y = log₁₀ x，则 dy/dx = 1/(x ln 10)。考试中可能要求对 log₂ x 等求导。
5. Chain Rule with Exponential and Logarithmic Functions | 指数与对数函数的链式法则
exponent or argument is a function more complex than a linear term. For eᵘ, dy/dx = eᵘ × u'; for ln u, dy/dx = u'/u.
当指数或真数为非线性函数时，常用链式法则。对于 eᵘ，dy/dx = eᵘ × u'；对于 ln u，dy/dx = u'/u。
Worked example: Differentiate y = ex². Let u = x², u' = 2x, so dy/dx = eˣ² × 2x = 2x eˣ². Similarly, for y = ln(sin x), u = sin x, u' = cos x, derivative = cos x / sin x = cot x.
示例：求 y = ex² 的导数。设 u = x²，u' = 2x，则 dy/dx = eˣ² × 2x = 2x eˣ²。类似地，求 y = ln(sin x) 的导数，设 u = sin x，u' = cos x，得 cos x / sin x = cot x。
6. Product Rule Involving Exponentials and Logs | 含指数与对数的乘法法则
t of an exponential or logarithmic part and another function, apply the product rule: (uv)' = u'v + uv'. For example, y = x eˣ. Let u = x, v = eˣ, then u' = 1, v' = eˣ, so dy/dx = 1·eˣ + x·eˣ = eˣ(1 + x).
当函数为指数或对数部分与另一函数相乘时，使用乘法法则：(uv)' = u'v + uv'。例如 y = x eˣ，设 u = x，v = eˣ，则 u' = 1，v' = eˣ，因此 dy/dx = eˣ + x eˣ = eˣ(1+x)。
Another common example: y = x² ln x. u = x², v = ln x, u' = 2x, v' = 1/x. Result: 2x ln x + x²·(1/x) = 2x ln x + x = x(2 ln x + 1).
另一个常见例子：y = x² ln x。u = x²，v = ln x，u' = 2x，v' = 1/x。结果为 2x ln x + x²·(1/x) = 2x ln x + x = x(2 ln x + 1)。
7. Quotient Rule with Exponentials and Logs | 含指数与对数的除法法则
The quotient rule (u/v)' = (u'v - uv') / v² is used when division is present. For instance, y = eˣ / x. u = eˣ, v = x, u' = eˣ, v' = 1, so dy/dx = (eˣ·x - eˣ·1) / x² = eˣ(x-1)/x².
除法法则 (u/v)' = (u'v - uv') / v² 应用于分式函数。例如 y = eˣ / x，u = eˣ，v = x，u' = eˣ，v' = 1，则 dy/dx = (eˣ·x - eˣ·1)/x² = eˣ(x-1)/x²。
n x·1) / x² = (1 - ln x) / x². Quotient rule can sometimes be avoided by rewriting, but it's essential to master.
再如 y = ln x / x，dy/dx = ((1/x)·x - ln x·1) / x² = (1 - ln x)/x²。有时可改写函数避免除法法则，但必须熟练掌握。
8. Implicit Differentiation Involving Exponentials and Logs | 含指数与对数的隐函数求导
When equations link x and y implicitly, and terms like eʸ or ln y appear, differentiate with respect to x, treating y as a function of x. For eʸ, derivative is eʸ·dy/dx. For ln y, it is (1/y)·dy/dx.
当方程以隐式关联 x 和 y，且含有 eʸ 或 ln y 等项时，对 x 求导并将 y 视作 x 的函数。eʸ 的导数为 eʸ·dy/dx；ln y 的导数为 (1/y)·dy/dx。
y(cos x - 2x). Implicit differentiation techniques are crucial for related rates and tangent problems.
示例：x² + ln y = sin x。求导：2x + (1/y)·dy/dx = cos x，因此 dy/dx = y(cos x - 2x)。隐函数求导技术对相关变化率和切线问题至关重要。
9. Logarithmic Differentiation | 对数求导法
] or products/quotients. Take natural log of both sides, expand using log laws, then differentiate implicitly.
对数求导法是处理 y = [f(x)]^[g(x)] 形式或连乘积、商的有力工具。两边取自然对数，利用对数性质展开，再隐式求导。
Example: y = xˣ ln x. Differentiate: (1/y) dy/dx = ln x + x·(1/x) = ln x + 1, so dy/dx = y (ln x + 1) = xˣ (ln x + 1). This method is often examined in Edexcel papers.
例题：y = xˣ。取 ln：ln y = x ln x。求导：(1/y) dy/dx = ln x + x·(1/x) = ln x + 1，所以 dy/dx = y (ln x + 1) = xˣ (ln x + 1)。此方法在 Edexcel 试卷中常考。
10. Applications and Exam Tips | 应用与应试技巧
Questions often combine these rules to differentiate composite functions, find stationary points, or solve differential equations. Always simplify expressions first where possible, and be careful with the domain: ln x requires x > 0, and eˣ is always positive. In Edexcel A-Level exams, marks are awarded for correct application of the chain, product, and quotient rules. Practice past paper problems involving eˣ and ln x to build fluency.
题目常综合运用这些法则求复合函数导数、驻点或解微分方程。尽可能先化简表达式，并注意定义域：ln x 要求 x>0，eˣ 恒正。在 Edexcel A-Level 考试中，正确运用链式、乘法和除法法则可得步骤分。多做涉及 eˣ 和 ln x 的历年真题以提升熟练度。
Key formulas to memorize: d/dx(eˣ)=eˣ; d/dx(eᵏˣ)=k eᵏˣ; d/dx(ln x)=1/x; d/dx(aˣ)=aˣ ln a; d/dx(logₐ x)=1/(x ln a). Keep a resource sheet during revision.
必须牢记的核心公式：d/dx(eˣ)=eˣ；d/dx(eᵏˣ)=k eᵏˣ；d/dx(ln x)=1/x；d/dx(aˣ)=aˣ ln a；d/dx(logₐ x)=1/(x ln a)。复习时可制作公式卡片。
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