What is the Laplace transform of $$f(t)=\frac{d^{n}}{d t^{n}} \delta(t)$$, where $\delta(t)$ is the Dirac-Delta distribution.
I am really not sure, but i am expecting that by using the formula: $$\mathcal{L}\left\{f^{(n)}\right\}=s^{n} \mathcal{L}\{f\}-s^{n-1} f(0)-s^{n-2} f^{\prime}(0)-s^{n-3} f^{\prime \prime}(0)-\ldots-f^{(n-1)}(0)$$, we have $$\mathcal{L}\left(\frac{d^{n}}{d t^{n}} \delta(t)\right)=s^n(1)-0-0-0...-0=s^n$$
But Mathematically how to prove formally that all the derivatives are zeros?