the transform of a derivative

This is the single rule that makes the whole Laplace method worth learning. Everything else is supporting cast; this is the star. It says that the operation we most dread in a differential equation — taking a derivative — turns, in the s-domain, into something almost trivial: multiplying by s. That is the magic that converts calculus into algebra.

The first-derivative rule is L{f'(t)} = s F(s) - f(0). In words: the transform of the derivative is s times the transform of the function, minus the starting value f(0). The second-derivative rule builds on it: L{f''(t)} = s^2 F(s) - s f(0) - f'(0). Notice two things. First, each derivative roughly brings down a factor of s out front — that is the 'differentiation becomes multiplication' effect. Second, and crucially, the initial values f(0), f'(0) are subtracted off automatically. They are not an afterthought you bolt on at the end; they enter the equation the moment you transform. This rule comes from applying integration by parts to the defining integral, where the e^(-st) factor at the boundaries produces exactly the f(0) term.

Because of this rule, a differential equation in y(t) — full of y'' and y' — becomes an algebraic equation in Y(s) with the numbers f(0), f'(0) already plugged in. There is no separate step where you 'apply the initial conditions' to pin down arbitrary constants; the Laplace method never produces arbitrary constants in the first place. This is the deep advantage of the method, and this rule is precisely where that advantage is born.