Explanation of #30 in the Laplace Transform Table

Several of the entries in the table involve products, such as #9 through #14 and #19 through #23.

#30 involves the product of a power of t and a function, f(t).

It says that
1) if we have to find the Laplace transform of a function which is the same as
(tⁿ times a second function) and
2) if we can find the transform of that second function by itself,
then we can get the transform of the first function by taking the n-th derivative of (the Laplace transform of the second function) and multiplying by the n-th power of (-1).

So the steps needed are to find the transform of the second function and then to take its derivative.