---
title: "Laplace transform of derivative/shift property"
subject: "Mathematics III"
chapter: 3
key: "semester_3/mathematics_iii:laplace-transform-of-derivative-shift-property-two"
source: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transform-of-derivative-shift-property-two
site: BCT Engineering
---

# Laplace transform of derivative/shift property

Mathematics III · Chapter 3: Laplace Transform

- Asked in: 2067 Magh
- Video: [LAPLACE TRANSFORMS: SHIFTING THEOREM | shifting property|](https://www.youtube.com/watch?v=tS3tIkSURnA) — TUTOR THINK 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/mathematics_iii/laplace-transform-of-derivative-shift-property-two
- Whole chapter: https://www.bctengineer.com/md/semester_3/mathematics_iii/chapter-3

## Question

If L{f(t)} = F(s), then prove L{e^(at) f(t)} = F(s − a).

## Short answer

By definition, L{e^(at) f(t)} = ∫₀^∞ e^(−(s−a)t) f(t) dt = F(s−a). This is the first shifting theorem, valid for s−a > k where f is of exponential order k.

## Answer

**First shifting theorem.** If L{f(t)} = F(s), then L{e^(at) f(t)} = F(s − a).

**Proof.** By definition of the Laplace transform,

```text
L{e^(at) f(t)} = ∫₀^∞ e^(−st) e^(at) f(t) dt
               = ∫₀^∞ e^(−(s − a)t) f(t) dt
```

The last integral is the same integral that defines F(s), with s replaced by (s − a). Hence

```text
L{e^(at) f(t)} = F(s − a)
```

This is valid for s − a > k, where f is of exponential order k (|f(t)| ≤ M e^(kt)), so that F(s − a) exists. ∎

**Examples.**
- L{e^(at)} = 1/(s − a), since L{1} = 1/s.
- L{e^(−2t) cos 3t} = (s + 2)/((s + 2)² + 9), since L{cos 3t} = s/(s² + 9).
- L{e^(3t) t²} = 2/(s − 3)³.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The proof of the first shifting theorem begins with
   A. The definition integral of e^(-st) e^(at) f(t)
   B. Integration by parts
   C. The convolution integral
   D. The Taylor series of e^(at)
   Answer: A. Combine the exponentials. (Key point: Starting point)

2. e^(-st) e^(at) equals
   A. e^(-(s+a)t)
   B. e^(-(s-a)t)
   C. e^(-st/a)
   D. e^(-sat)
   Answer: B. Exponents add: -st + at. (Key point: Key step)

3. F(s - a) exists when
   A. s < a
   B. s > 0 only
   C. s - a > k, where f has exponential order k
   D. a > k
   Answer: C. F(s) exists for s > k. (Key point: Condition)

4. L{e^(-2t) cos 3t} =
   A. (s-2)/((s-2)^2 + 9)
   B. s/((s+2)^2 + 9)
   C. 3/((s+2)^2 + 9)
   D. (s+2)/((s+2)^2 + 9)
   Answer: D. Replace s by s + 2 in s/(s^2+9). (Key point: Example)

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