Time Differentiation Property of The Laplace Transform

Given that $ F(s) $ is the Laplace transform of $ f(t) $, the Laplace transform of its derivative is To integrate this by parts, we let $ u=e^{-s t}, d u=-s e^{-s t} d t $, and $ d v= $ $ (d f / d t) d t=d f(t), v=f(t) $. Then or The Laplace transform of the second derivative of $ f(t) $ is a repeated application of Eq. (1) as Continuing in this manner, we can obtain the Laplace transform of the $ n $th derivative of $ f(t) $ as As an example, we can use Eq. (1) to obtain the Laplace transform of the sine from that of the cosine. If we let $ f(t)=\cos \omega t $, then $ f(0)=1 $ and $ f^{\prime}(t)=-\omega \sin \omega t $. Using Eq. (1) and the scaling property, as expected.