# deriving inverse of Laplace transform

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?

• Are you try to get the inverse result for a particular function by the inverse theorem, or you want to prove the inverse theorem? As the proof of the inverse theorem is never about algebraic operations, it's very involving. – Yujie Zha Apr 29 '17 at 13:19
• I'm trying to derive the general formula for the inverse of the laplace transform. – user56834 Apr 29 '17 at 13:28
• Maybe you could check out this doc – Yujie Zha Apr 29 '17 at 13:32