How did this function turn to the other through the laplace transform?

Question

I was studying for my exam doing exercises, and in one of the questions where you had to use the laplace transform, they transformed this: $$X'(t)=AX(t)$$ into this:$$sX(s)-X(t=0)=AX(s)$$ where $X(s)=L\{X(t)\}$
Could someone explain to me the left side of the second function please? I've been looking at it for an hour like an idiot and cant figure it out...

Answer 1

Following on from my comment:

The left-hand side, namely $sX(s)-X(t=0)$, is the Laplace transform of the derivative $X'(t)$. You should be able to find it in most Laplace transform tables.

I'll provide a link to the 'proof' and use of it (as there is too much to possibly write here).

http://www.math.uah.edu/howell/DEtext/Part4/Laplace_derivatives.pdf