You have to use the change event on the input itself if you want to respond to manual input, because the changeDate event is only for when the date is changed using the datepicker.
Try something like this:
$(document).ready(function() {
$('.datepicker').datepicker({
format: "yyyy-mm-dd",
})
//Listen for the change even on the input
.change(dateChanged)
.on('changeDate', dateChanged);
});
function dateChanged(ev) {
$(this).datepicker('hide');
if ($('#startdate').val() != '' && $('#enddate').val() != '') {
$('#period').text(diffInDays() + ' d.');
} else {
$('#period').text("-");
}
}