III.1.4 Laplace Transform of derivatives

The Laplace transform of derivative of a function f(t) is derived in the following manner:

(III.19)

Where f(0) is the value of the function at t=0. Similarly it can be proved that

(III.20)

Where f'(0) is the value of the derivative of the function at t=0. In general, it can be proved that

(III.21)

Where are the initial conditions of the respective-order derivatives of the function.



III.1.5 Laplace Transform of Integrals

The Laplace transform of integral of a function f(t) is derived in the following manner:

(III.22)