quantum mechanics - Squaring the momentum operator in QM becomes a second derivative. How?

$\frac{p^2}{2m}$ is the Kinetic energy in classical mechanics. However, the same $p^2$ becomes the second derivative $\frac{\partial ^2}{\partial x^2}$ in the Kinetic Energy operator in QM. I mean it is strange that a squaring process of a differential in an operator becomes the double differential. How was it deduced in the first place?