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The differentiation theorem for Laplace transforms states that

$\displaystyle {\dot x}(t) \leftrightarrow s X(s) - x(0),

where $ {\dot x}(t) \isdef \frac{d}{dt}x(t)$, and $ x(t)$ is any differentiable function that approaches zero as $ t$ goes to infinity. In operator notation,

$\displaystyle \zbox {{\cal L}_{s}\{{\dot x}\} = s X(s) - x(0).}

Proof: This follows immediately from integration by parts:
{\cal L}_{s}\{{\dot x}\} &\isdef & \int_{0}^\infty {\dot x}(t)...
...y} -
\int_{0}^\infty x(t) (-s)e^{-s t} dt\\
&=& s X(s) - x(0)
since $ x(\infty)=0$ by assumption.
Corollary: Integration Theorem

$\displaystyle \zbox {{\cal L}_{s}\left\{\int_0^t x(\tau)d\tau\right\} = \frac{X(s)}{s}}

Thus, successive time derivatives correspond to successively higher powers of $ s$, and successive integrals with respect to time correspond to successively higher powers of $ 1/s$.
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