Here we consider not a causal function
directly but its derivatives
, …(which are also causal.) The Laplace transform of derivatives will be invaluable when we apply the Laplace transform to the solution of constant coefficient ordinary differential equations.
then we shall seek an expression for
in terms of the function
Now, by the definition of the Laplace transform
This integral can be simplified using integration by parts:
(As usual, we assume that contributions arising from the upper limit,
, are zero.) The integral on the right-hand side is precisely the Laplace transform of
which we naturally replace by
As an example, we know that if
and so, according to the result just obtained,
a result we know to be true.
We can find the Laplace transform of the second derivative in a similar way to find:
(The reader might wish to derive this result.) Here
is the derivative of