Laplace Transform of Derivative/Examples/Example 1

Examples of Use of Laplace Transform of Derivative

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

$\laptrans {-3 \sin 3 t} = \dfrac {-9} {s^2 + 9}$

Proof

\(\ds \laptrans {-3 \sin 3 t}\)

\(=\)

\(\ds s \laptrans {\cos 3 t} - \cos 0\)

Laplace Transform of Derivative, Derivative of $\cos a x$

\(\ds \)

\(=\)

\(\ds s \dfrac s {s^2 + 9} - 1\)

Laplace Transform of Cosine, Cosine of Zero is One

\(\ds \)

\(=\)

\(\ds \dfrac {-9} {s^2 + 9}\)

simplification

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $5$. Laplace transform of derivatives