Derivatives must vanish near a point in any coordinate system These differentiable manifold notes of Nigel Hitchin argue (on page 16) that

"on a compact manifold any function has a maximum, and in any
coordinate system in a neighbourhood of that point, its derivative
must vanish".

Why is this (that any coordinate system must have this zero derivative property at a point)? Does this follow from an argument similar to the chain rule? That when the derivative vanishes in one system, any transformation gets zeroed by these derivatives being zero? It seems difficult to imagine how no coordinate change could alter the derivative, but perhaps it is obvious.
 A: An answer to the comment asking for a

way to explain what a change of coordinates actually is, intuitively?

that may also help with the question.
Consider all the paper maps you know of the globe (stay away from the poles for this discussion). On each you have the  the circles of longitude and latitude - which will not be circles. The usual rectangular coordinate system on the rectangular map tells you how to identify points on the globe by their map $(x,y)$ coordinates, so you have a coordinate system for a patch of the globe. Different map projections correspond to different local coordinate systems.
Converting coordinates of a point on the globe from those on one map to those on another is a change of coordinates.
Now think about the height above sea level as a function defined on the globe. It will have a local maximum (in fact, a global maximum) at the top of Mt Everest. If you calculate heights as a function $f$ of $(x,y)$ on any map (in any coordinate system) that function will have a local maximum and horizontal tangent plane at the point corresponding to Mt Everest.
This example suggests the standard vocabulary that describes  manifold structure as an atlas of charts.
