Let $f \in \mathcal { C } _ { b } ^ { 0 } ( [ 0 , + \infty [ )$. We define the Laplace transform of $f$ by the function
$$\mathcal { L } ( f ) : t \in ] 0 , + \infty \left[ \mapsto \int _ { 0 } ^ { + \infty } e ^ { - t x } f ( x ) d x \right.$$
Prove that $\mathcal { L } ( f )$ is well-defined and of class $\mathcal { C } ^ { 1 }$ on $] 0 , + \infty [$, and express its derivative.