Laplace differentiation is made as 

Laplace[ df(t)/dt ] = s F(s) 

But in practice matlab and other software use

Laplace[ df(t)/dt ] = F(s) *  s / (tf s + 1) 

Simple way to solve this problem is                                          Check http://math.stackexchange.com/questions/357391/inverse-laplace-transform-of-s-s1

=ss+a=(s+a)−as+a=s+as+a−as+a=1−as+a=δ(t)−(ae−at)