📚 Laplace Transforms of Derivatives | 导数的拉普拉斯变换
The Laplace transform is a cornerstone technique for solving linear differential equations. Its ability to turn derivatives into algebraic expressions involving initial conditions makes it an indispensable tool in engineering and physics. This article covers the key theorems, derivations, and practical uses of the Laplace transforms of derivatives, designed to support IB Mathematics Higher Level students in mastering the topic.
拉普拉斯变换是求解线性微分方程的基石技术。它能将导数转化为包含初始条件的代数表达式，使其成为工程和物理中不可或缺的工具。本文涵盖了导数拉普拉斯变换的关键定理、推导与实际应用，旨在帮助 IB 数学高水平学生掌握这一主题。
1. The Derivative Theorem and Its Importance | 导数定理及其重要性
sform of its first derivative introduces a multiplication by s and subtracts the initial value f(0). This simple connection is the foundation for transforming entire differential equations into algebraic problems.
如果函数 f(t) 的拉普拉斯变换为 F(s) = L{f(t)}，那么其一阶导数的变换会引入与 s 的乘积并减去初始值 f(0)。这一简单联系是将微分方程整体转换为代数问题的基础。
∞) and of exponential order, which covers nearly all functions encountered in IB problems.
该性质对所有在 [0, ∞) 上连续且具有指数阶的函数成立，这几乎涵盖了 IB 问题中遇到的所有函数。
2. Derivation by Integration by Parts | 由分部积分法推导
Start from the definition: L{f '(t)} = ∫₀∞ e⁻ˢᵗ f '(t) dt. Set u = e⁻ˢᵗ, dv = f '(t) dt. Then du = –s e⁻ˢᵗ dt and v = f(t). Integration by parts gives [e⁻ˢᵗ f(t)]₀∞ – ∫₀∞ (–s e⁻ˢᵗ) f(t) dt.
从定义出发：L{f '(t)} = ∫₀∞ e⁻ˢᵗ f '(t) dt。设 u = e⁻ˢᵗ，dv = f '(t) dt，则 du = –s e⁻ˢᵗ dt，v = f(t)。分部积分得到 [e⁻ˢᵗ f(t)]₀∞ – ∫₀∞ (–s e⁻ˢᵗ) f(t) dt。
ming the function is of exponential order, the boundary term vanishes at infinity, leaving –f(0) + s ∫₀∞ e⁻ˢᵗ f(t) dt = sF(s) – f(0).
假设函数具有指数阶，无穷远处的边界项消失，剩下 –f(0) + s ∫₀∞ e⁻ˢᵗ f(t) dt = sF(s) – f(0)。
L{f '(t)} = sF(s) – f(0)
3. The First Derivative Formula in Practice | 一阶导数公式的实际运用
The relation L{f '(t)} = sF(s) – on is zero, this simplifies to pure multiplication by s, which is analogous to the operator D in calculus being replaced by s.
关系式 L{f '(t)} = sF(s) – f(0) 告诉我们，求一次时间导数相当于将变换乘以 s 再减去初始值。如果初始条件为零，则简化为纯粹乘以 s，这类似于微积分中的算子 D 被 s 替代。
This formula is directly used when analysing simple RC circuits, radioactive decay, and first‑order reaction kinetics in the Laplace domain.
该公式在分析简单 RC 电路、放射性衰变以及一阶反应动力学时可直接用于拉普拉斯域。
4. Transform of the Second Derivative | 二阶导数的变换
Applying the first‑derivative rule twice yields the second‑derivative transform. Let g(t) = f '(t); then L{f ''(t)} = L{g'(t)} = sG(s) – g(0) = s[sF(s) – f(0)] – f '(0).
连续两次应用一阶导数规则即得二阶导数变换。令 g(t) = f '(t)，则 L{f ''(t)} = L{g'(t)} = sG(s) – g(0) = s[sF(s) – f(0)] – f '(0)。
L{f ''(t)} = s²F(s) – s f(0) – f '(0)
Notice that both the function' at t = 0 appear. The polynomial coefficient of F(s) reflects the order of differentiation.
注意函数值及其在 t = 0 处的斜率都出现了。F(s) 的多项式系数反映了微分的阶数。
5. General Rule for the n‑th Derivative | n 阶导数的一般规则
By mathematical induction, the pattern extends to any order n. The Laplace transform of the n‑th derivative is:
通过数学归纳法，这一模式可推广至任意 n 阶。n 阶导数的拉普拉斯变换为：
