multiplication by t (differentiation in s)

There is a pleasing symmetry hiding in the Laplace world. The derivative rule showed that differentiating in t corresponds (roughly) to multiplying by s. This rule is the partner on the other side: multiplying by t in the time domain corresponds to differentiating in s. The two operations trade places when you cross between domains.

The rule is L{t f(t)} = -F'(s), where F'(s) means the derivative of F with respect to s. More generally, multiplying by t^n brings down n derivatives with a sign: L{t^n f(t)} = (-1)^n F^(n)(s). The minus sign comes from the e^(-st) inside the integral: differentiating that factor with respect to s pulls down a -t, so each derivative in s is the same as one multiplication by -t. This gives you a way to build new transforms from old ones without doing a fresh integral — you just differentiate the transform you already have.

It is also the cleanest source of the power transforms: starting from L{1} = 1/s and differentiating repeatedly recovers L{t} = 1/s^2, L{t^2} = 2/s^3, and the whole family. More importantly, it is the rule you need when a forcing term has the form t times something — for example t sin(t) or t e^(t), which appear precisely in the resonant cases where the usual guess must be multiplied by t. When you meet such a term, differentiation in s is the tool that transforms it cleanly.