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okay so what im going to try doing now is um the last tool that we need before we can actually start um using laplace very interesting things like solving um differential equations set up as a linear system of equations um and and to do that all we need to do is look at the the final laplace transform um of uh the derivative functions so things like um the first derivative of f prime t f double prime t f triple prime t um as such um and just to give you a heads up that the algebra is extremely extremely easy and extremely easy to get wrong um its very very fiddly um and ive got it wrong nearly every single time just by carelessness so its uh youll probably do it a lot better than me but um just uh just a brief warning so lets lets just dive in and uh and have a look at the other functions that we need to uh look at so uh what we wanna do is find the laplace transform of f prime t okay so thats the first derivative of t yeah um so as we know by now thats going to be the semi-i
