Use the relation of Laplace Transform and its derivative to figure out $L\left\{t\right\}$，$L\left\{t^2\right\}$，$L\left\{t^n\right\}$

Question

If $F(s) = L\left\{f(t)\right\}$, then $F'(s) = -L\left\{tf(t)\right\}$

Use this relation to determine

$(a)$ $L\left\{t\right\}$

$(b)$ $L\left\{t^2\right\}$

$(c)$ $L\left\{t^n\right\}$ for any positive integer $n$.

I was able use the definition of Laplace Transform and integration to figure out $(a)$,$(b)$ and $(c)$.
Namely

$L\left\{t\right\} = \dfrac{1}{s^2}$

$L\left\{t^2\right\}=\dfrac{2}{s^3}$

$L\left\{t^n\right\} = \dfrac{n!}{s^{n+1}}$

But how I use the relation between Laplace Transform and its derivative to figure out $(a)$,$(b)$ and $(c)$?

Answer 1

Induction. You know how to write down $F'(s)$. Now, $\mathscr L\{t\cdot t^n\}=\cdots$?

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For example, $\mathscr L\{t^3\}=\mathscr L\{t\cdot t^2\}=\mathscr L\{tf(t)\} $ with $f(t)=t^2$. If you knew that $\mathscr L\{t^2\}=\dfrac{2!}{s^3}$ then $\mathscr L\{t^3\}=-F'(s)=-\dfrac{d}{ds}\left(\dfrac{2!}{s^3}\right)=\dfrac{3!}{s^4}$.