Laplace Transform of Derivatives

For first-order derivative:
L{f(t)}=sL{f(t)}f(0)
 

For second-order derivative:
L{f(t)}=s2L{f(t)}sf(0)f(0)
 

For third-order derivative:
L{f(t)}=s3L{f(t)}s2f(0)sf(0)f(0)
 

For nth order derivative:

L{fn(t)}=snL{f(t)}sn1f(0)sn2f(0)fn1(0)

 

Proof of Laplace Transform of Derivatives
L{f(t)}=0estf(t)dt
 

Using integration by parts,
u=est
du=sestdt

dv=f(t)dt
v=f(t)
 

Thus,
L{f(t)}=[estf(t)]00f(t)(sestdt)

L{f(t)}=[f(t)est]0+s0estf(t)dt

L{f(t)}=[f(t)est]0+sL{f(t)}
 

Apply the limits from 0 to :
L{f(t)}=[f()ef(0)e0]+sL{f(t)}

L{f(t)}=f(0)+sL{f(t)}

L{f(t)}=sL{f(t)}f(0)           okay