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 nth 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.