laplace transform two forms

Question

$$L[f'''(t)]=\int e^{-st}f'(t)dt=L[f'(t)]$$

im not sure if this is how it works. please advice

Answer 1

It is easy to see that since

$$\frac{d}{ds}\int_0^\infty f'(t)e^{-st}\,dt=-\int_0^\infty tf'(t)e^{-st}\,dt$$

we have

$$\begin{align} \int_0^\infty t^nf'(t)e^{-st}\,dt&=(-1)^n\frac{d^n}{ds^n}\int_0^\infty f'(t)e^{-st}\,dt\\\\ &=(-1)^n\frac{d^n}{ds^n}\left(sF(s)-f(0)\right)\\\\ &=(-1)^n\frac{d^n(sF(s))}{ds^n}\\\\ &=(-1)^n sF^{(n)}(s)+(-1)^n F^{(n-1)}(s) \end{align}$$