The concavity of a function is described by its second derivative.
-The expression will not directly tell us whether the function is concave up or concave down and where it changes, though graphing the function may give us some sort of idea.
-The first derivative tells us the critical/important values (where there could be asymptotes), as well as where the function is increasing/decreasing.
-The second derivative tells us again the critical/important values, but also where the function is concave up/down. We find concavity in the same way we find increasing/decreasing using the first derivative.
-The third derivative, at least in my experience, has only been used to find acceleration of a function.
Hope this helps!! :)
