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HTML

<div class="box">
      s<br>
      s<br>
      s<br>
      s<br>
      s<br>
</div>

<button class="btn">
  add row
</button>

<script src="https://ajax.googleapis.com/ajax/libs/jquery/3.1.0/jquery.min.js"></script>

CSS

.box {
  width: 250px;
  height: 150px;
  color: white;
  background-color: #1d1d1b;
  overflow: scroll;
}

JavaScript

$('.box').bind("DOMSubtreeModified propertychange", function(e) {
		$(this).animate({ scrollTop: $(this).prop("scrollHeight") }, 100);
});

$(document).on('click', '.btn', function(){
		$('.box').append('s<br>');
});