Updating DOM after AJAX inserts updated element

I have the following ajax call:

    $("#searchField").on('keyup', function(){
             var value = $(this).val();
              $.ajax({
                type : 'get',
                url : 'livesearch',
                data : {
                    search:value

                },
                success : function(data){
                  console.log(data);
                  $('#myDiv').html(data);
                }

              });

      });

It updates the myDiv element but DOM does not change.

How do I update the DOM?