A 39-minute DL-DL connection at LAX is fine, with the usual caveat that the flights are on time. If you miss your flight and there's only one or 2 flights a day -- or those flights are full -- you might end up losing 1/2 or 1 day of your vacation.

Before accepting this change, you might want to find out when the next flight is and how full it is, just to know how inconvenient a misconnect would be.