# Laplace Transform of Derivative

Cuthbert Nyack

To find the Laplace transform of a derivative, integrate the expression for
the definition of the Laplace Transform by parts to obtain:-
Evaluating the limits and multiplying by s gives the following:-
The above equation is usually rearranged and expressed as follows giving
the Laplace transform of f'(t) as a function of the Laplace Transform
of f(t):-
If f(t) in the above equation is replaced by f'(t), then the Laplace
Transform of the second derivative is obtained and shown below.

This can be continued for higher order derivatives and gives the
following expression for the Laplace Transform of the n^{th}
derivative of f(t).

The Laplace transform of the integral of f(t) can be shown to be given by:-

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Copyright 1996 © Cuthbert A. Nyack.