When you change the input date it updates the text area. It does this by replacing whatever is in the text area with your update - so it disappears.
What you need to do is update the text are with a combination of what was in it and your new value from the date update. This means changing your date-insertion script.
something like:
textObj =document.getElementbyId('startDate');
textObj.value = textObj.value + ' ' + the_new_start_date;