Differential: Smoothness

Question

This is a lemma for another thread.

Given smooth manifolds.

The differential is a smooth map: $$F:M\to N:\quad F\text{ smooth}\implies\mathrm{d}F\text{ smooth}$$ How to check this?

Answer 1

This answer is community wiki.

Regard its coordinate representation: $$\widehat{\mathrm{d}F}(\hat{x})v=(\hat{F}(\hat{x}),\mathrm{J}\hat{F}(\hat{x})v)$$ This is clearly smooth.