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

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

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 $\mathscr L\{t^3\}=-F'(s)=-\dfrac{d}{ds}\left(\dfrac{2!}{s^3}\right)=\dfrac{3!}{s^4}$. 
