However, if I read your post correctly, I think LaplaceTransform is working correctly, in that the Laplace transform of 1 is 1/s.

In[1]:= LaplaceTransform[g'[t], t, s]

Out[1]= -g[0] + s LaplaceTransform[g[t], t, s]

In[2]:= LaplaceTransform[g'[x], t, s]

Out[2]= Derivative[1][g][x]/s

In[3]:= LaplaceTransform[1, t, s]

Out[3]= 1/s

Thanks I will learn how to post this. I started to realize that in order to explain this one I have a bigger problem than just mathematica. I think the person would also have to understand green's functions. I can do them by hand but not with Mathematica and the answer to the problem is easy if you understand green's functions. I do not know if I should go to the mathematics site or the physics site. You have f'[x]e^xs, you can get f[x] by integration by parts. It puts the derivative with respect to x on the e^sx and you get f[x] by itself. So you get -sf[x]e^xs so you now have f[x] by itself, with no derivatives on it. If you have f'[x]e^ys you cannot do an integration by parts, since the derivative of e^ys with respect to x is zero. So the laplace transform e^ys just stays there. If I can get someone to understand this is where the problem is, maybe they can help me. Mathematica should leave it alone, since you do not want to do a integration by parts, which gives 0. But it leaves (f'[x]e^ys)/s...[I will call a] and it is a problem with mathematica I do not understand. In other-words you have done nothing to the function or the exponential but mathematica puts in 1/s and nothing has been done to that part of the equation. David can you lead me in the right direction. It is like asking mathematica not to take a derivative by it gives and extra x or what ever variable.

I need to learn to do post's better because I cannot explain myself, until I can show the equations. The same thing would be true for a fourier transform. I can do it by hand but not with mathematica. I should not get the 1/s, if the variable is not t. It is like the partial derivative I did above where you hold y constant. You can take a laplace transform with respect to another variable and it has no effect but to leave the laplace transform. In order to do the laplace transforms you have to do a integration by parts, but if the variable is not the one you are taking the derivative of, you just leave it and do not do the integration by parts. I will have to learn the editor better. I could explain it so easy if I could just write out equations. Like I said I can do it by hand, but not mathematica, I do not know where the 1/s comes from, it should not be there. It would be like taking the derivative with respect to x of e^2y, it is zero. I cannot use this editor so I am having real problems communicating. I am really sorry.