I am trying to derive the inverse of the laplace transform myself, but I'm immediately stuck.
The laplace transform is: $$\mathcal L[f(t)](s)=\int_0^\infty e^{-st}f(t)dt$$
Intuitively, my first attempt would be to get rid of the integral with respect to $t$, by taking the derivative: $$\frac{d \mathcal L[f(t)](s)}{dt}=e^{-st}f(t), \text{or something...}$$
But this doesn't work, because the integral is not a function of $t$, since it is definite.
Can someone give me a hint on how to solve it, or a possible misconception that I may have?