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Derivatives of Exponential and Logarithmic Functions: Exercise 6E.3 Full Guide | 指数与对数函数的导数：练习6E.3 完整指南

2026-08-02 22:01 · 本文部分内容由AI辅助创作

📚 Derivatives of Exponential and Logarithmic Functions: Exercise 6E.3 Full Guide | 指数与对数函数的导数：练习6E.3 完整指南

In IB Mathematics, particularly in the Analysis and Approaches course, mastering differentiation techniques for exponential and logarithmic functions is crucial. Exercise 6E.3 typically focuses on applying the chain rule, product rule, and quotient rule to functions involving eˣ, aˣ, ln x, and logₐ x. This article will guide you through the key concepts, worked examples, and common pitfalls associated with this exercise set, helping you build confidence for both internal assessments and final examinations.

在 IB 数学，特别是分析与方法课程中，掌握指数与对数函数的微分技巧至关重要。练习 6E.3 通常侧重于将链式法则、乘积法则和商法则应用于涉及 eˣ、aˣ、ln x 和 logₐ x 的函数。本文将通过关键概念、范例和常见错误，带你深入理解这组练习，帮助你在内部评估与最终考试中建立信心。

1. Recap of Basic Derivatives | 基础导数回顾

Before tackling the problems in Exercise 6E.3, ensure you are completely comfortable with the basic derivatives of exponential and logarithmic functions. The derivative of eˣ is eˣ itself, which is a unique and powerful property. For a general exponential function aˣ (with a > 0, a ≠ 1), the derivative is aˣ ln a. Similarly, the derivative of ln x is 1/x for x > 0, and for logₐ x, it is 1/(x ln a). These foundational results must be memorised because they appear repeatedly in more complex differentiations.

在解决练习 6E.3 的问题之前，确保你完全熟悉指数和对数函数的基本导数。eˣ 的导数就是 eˣ 本身，这是一个独特且强大的性质。对于一般的指数函数 aˣ（a > 0，a ≠ 1），导数为 aˣ ln a。类似地，ln x 的导数是 1/x（x > 0），而 logₐ x 的导数为 1/(x ln a)。这些基础结论必须牢记，因为它们在更复杂的微分中会反复出现。

Function f(x)Derivative f'(x)Domain / Condition
eˣeˣx ∈ ℝ
aˣaˣ ln aa > 0, a ≠ 1
ln x1/xx > 0
logₐ x1/(x ln a)x > 0, a > 0, a ≠ 1

You should also remember that the derivative of ln|x| is 1/x for x ≠ 0. This extended form is useful when the argument of the logarithm can be negative, such as ln|2x - 1|. In IB examinations, a common first step is to rewrite roots and reciprocals as powers before differentiating.

你还应记住，ln|x| 的导数是 1/x（x ≠ 0）。当对数的自变量可能为负时，例如 ln|2x - 1|，这一扩展形式非常有用。在 IB 考试中，常见的第一步是把根式和倒数改写为幂的形式，再进行微分。

2. Chain Rule with eᵘ and aᵘ | eᵘ 与 aᵘ 的链式法则

In Exercise 6E.3 you will frequently encounter functions like e²ˣ, esin x, or 5x²+1 dx. The most common mistake is forgetting to multiply by the derivative of the exponent. For example, the derivative of e³ˣ is 3e³ˣ, not e³ˣ.

在练习 6E.3 中，你会经常遇到如 e²ˣ、esin x 或 5x²+1 等函数。链式法则指出，若 y = eᵘ 且 u 是 x 的可微函数，则 dy/dx = eᵘ · du/dx。对于一般底数 a，若 y = aᵘ，则 dy/dx = aᵘ ln a · du/dx。最常见的错误是忘记乘以指数的导数。例如，e³ˣ 的导数是 3e³ˣ，而不是 e³ˣ。

dy/dx = eᵘ · du/dx

dy/dx = aᵘ ln a · du/dx

When the exponent is linear, such as 4x + 7, the inner derivative is simply 4. Thus d/dx[e4x+7] = 4e4x+7. When the exponent is quadratic, trigonometric, or logarithmic, the inner derivative requires more care. Always write u explicitly if you are prone to skipping steps.

当指数为线性函数时，例如 4x + 7，内层导数就是 4。因此 d/dx[e4x+7] = 4e4x+7。当指数为二次、三角或对数函数时，内层导数需要更小心。如果你容易跳步，就明确写出 u。

3. Chain Rule with ln u and logₐ u | ln u 与 logₐ u 的链式法则

For logarithmic functions, the chain rule gives d/dx[ln u] = (1/u) · du/dx = u' n(3x + 1) has derivative 3/(3x + 1). Similarly, y = ln(x² + 1) has derivative 2x/(x² + 1).

对于对数函数，链式法则给出 d/dx[ln u] = (1/u) · du/dx = u'/u。对于以 a 为底的对数，d/dx[logₐ u] = (1/(u ln a)) · du/dx。这些公式在练习 6E.3 中至关重要。例如，y = ln(3x + 1) 的导数是 3/(3x + 1)。类似地，y = ln(x² + 1) 的导数是 2x/(x² + 1)。

d/dx [ln u] = (1/u) · du/dx = u'/u

d/dx [logₐ u] = (1/(u ln a)) · du/dx

owever, this shortcut only works for positive k and positive x. In general, keep the chain rule structure until you are sure simplification is valid.

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当对数的自变量是常数倍时，会出现一个有用的简化：ln(kx) 的导数是 1/x，因为 ln(kx) = ln k + ln x，而 ln k 是常数。但这个捷径只对正数 k 和正数 x 成立。一般来说，在你确定简化有效之前，保留链式法则的结构。

4. Product Rule with Exponential and Logarithmic Functions | 指数与对数函数的乘积法则

nomials or trigonometric functions. The product rule states that if y = uv, then dy/dx = u'v + uv'. When one factor is eˣ, its derivative is again eˣ ctoring. When one factor is ln x, its derivative is 1/x, which may create a rational term.

许多练习 6E.3 的题目会把指数或对数函数与多项式或三角函数结合起来。乘积法则指出，若 y = uv，则 dy/dx = u'v + uv'。当其中一个因子是 eˣ 时，它的导数仍是 eˣ，这通常可以进行因式分解。当其中一个因子是 ln x 时，它的导数是 1/x，可能会产生有理项。

dy/dx = u'v + uv'

Always identify u and v clearly before differentiating. A common error is to differentiate the product as the product of derivatives, which is incorrect. The product rule must be used whenever two non-constant factors are multiplied.

在微分之前，一定要清楚地识别 u 和 v。一个常见错误是把乘积的导数当成导数的乘积，这是不正确的。只要两个非常数因子相乘，就必须使用乘积法则。

5. Quotient Rule and Mixed Expressions | 商法则与混合表达式

quotient rule is used when one function is divided by another. If y = u/v, then dy/dx = (u'v - uv')/v². Exercise 6E.3 often includes quotients such as eˣ ator of the quotient rule has a minus sign, and reversing the order is a frequent mistake.

当一个函数除以另一个函数时，使用商法则。若 y = u/v，则 dy/dx = (u'v - uv')/v²。练习 6E.3 经常包含 eˣ/x、ln x/x 或 (1 + ln x)/x 这样的商。商法则的分子中有一个减号，颠倒顺序是常见错误。

dy/dx = (u'v - uv') / v²

etimes a quotient can be rewritten as a product using a negative exponent, for example 1/eˣ = e-x. This can simplify differentiation, but you must still respect the chain rule. In IB exams, either method earns full marks if the algebra is correct.

有时商可以改写为使用负指数的乘积，例如 1/eˣ = e-x。这可以简化微分，但你仍必须遵守链式法则。在 IB 考试中，只要代数正确，两种方法都能得满分。

6. Logarithmic Differentiation for Complex Powers | 复杂幂函数的对数微分法

When both the base and the exponent depend on x, such as y = xˣ or y = (sin x)x (x). Then differentiate implicitly. This transforms a power into a product, which is much easier to handle.

当底数和指数都依赖于 x 时，例如 y = xˣ 或 y = (sin x)x，对数微分法是标准方法。对两边取自然对数：ln y = g(x) ln f(x)。然后进行隐式微分。这把幂转化为乘积，处理起来容易得多。

y = f(x)g(x) ⇒ ln y = g(x) ln f(x)

(1/y) dy/dx = g'(x) ln f(x) + g(x) f'(x)/f(x)

dy/dx = y [ g'(x) ln f(x) + g(x) f'(x)/f(x) ]