Correct Answer:

function addDays(date, days) {
  var result = new Date(date);
  result.setDate(result.getDate() + days);
  return result;
}

Incorrect Answer:

This answer sometimes provides the correct result but very often returns the wrong year and month. The only time this answer works is when the date that you are adding days to happens to have the current year and month.

// Don't do it this way!
function addDaysWRONG(date, days) {
  var result = new Date(); // not instatiated with date!!! DANGER
  result.setDate(date.getDate() + days);
  return result;
}

Proof / Example

Check this JsFiddle

// Correct
function addDays(date, days) {
    var result = new Date(date);
    result.setDate(result.getDate() + days);
    return result;
}

// Bad Year/Month
function addDaysWRONG(date, days) {
    var result = new Date();
    result.setDate(date.getDate() + days);
    return result;
}

// Bad during DST
function addDaysDstFail(date, days) {
    var dayms = (days * 24 * 60 * 60 * 1000);
    return new Date(date.getTime() + dayms);    
}

// TEST
function formatDate(date) {
    return (date.getMonth() + 1) + '/' + date.getDate() + '/' + date.getFullYear();
}

$('tbody tr td:first-child').each(function () {
    var (this);
    var ('<td/>').insertAfter($in).addClass("answer");
    var $outFail = $('<td/>').insertAfter(outDstFail = $('<td/>').insertAfter($outFail);
    var date = new Date($in.text());
    var correctDate = formatDate(addDays(date, 1));
    var failDate = formatDate(addDaysWRONG(date, 1));
    var failDstDate = formatDate(addDaysDstFail(date, 1));

    $out.text(correctDate);
    $outFail.text(failDate);
    $outDstFail.text(failDstDate);
    $outFail.addClass(correctDate == failDate ? "right" : "wrong");
    $outDstFail.addClass(correctDate == failDstDate ? "right" : "wrong");
});

body {
    font-size: 14px;
}

table {
    border-collapse:collapse;
}
table, td, th {
    border:1px solid black;
}
td {
    padding: 2px;
}

.wrong {
    color: red;
}
.right {
    color: green;
}
.answer {
    font-weight: bold;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/1.11.1/jquery.min.js"></script>
<table>
    <tbody>
        <tr>
            <th colspan="4">DST Dates</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>03/10/2013</td></tr>
        <tr><td>11/03/2013</td></tr>
        <tr><td>03/09/2014</td></tr>
        <tr><td>11/02/2014</td></tr>
        <tr><td>03/08/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr>
            <th colspan="4">2013</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2013</td></tr>
        <tr><td>02/01/2013</td></tr>
        <tr><td>03/01/2013</td></tr>
        <tr><td>04/01/2013</td></tr>
        <tr><td>05/01/2013</td></tr>
        <tr><td>06/01/2013</td></tr>
        <tr><td>07/01/2013</td></tr>
        <tr><td>08/01/2013</td></tr>
        <tr><td>09/01/2013</td></tr>
        <tr><td>10/01/2013</td></tr>
        <tr><td>11/01/2013</td></tr>
        <tr><td>12/01/2013</td></tr>
        <tr>
            <th colspan="4">2014</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2014</td></tr>
        <tr><td>02/01/2014</td></tr>
        <tr><td>03/01/2014</td></tr>
        <tr><td>04/01/2014</td></tr>
        <tr><td>05/01/2014</td></tr>
        <tr><td>06/01/2014</td></tr>
        <tr><td>07/01/2014</td></tr>
        <tr><td>08/01/2014</td></tr>
        <tr><td>09/01/2014</td></tr>
        <tr><td>10/01/2014</td></tr>
        <tr><td>11/01/2014</td></tr>
        <tr><td>12/01/2014</td></tr>
        <tr>
            <th colspan="4">2015</th>
        </tr>
        <tr>
            <th>Input</th>
            <th>+1 Day</th>
            <th>+1 Day Fail</th>
            <th>+1 Day DST Fail</th>
        </tr>
        <tr><td>01/01/2015</td></tr>
        <tr><td>02/01/2015</td></tr>
        <tr><td>03/01/2015</td></tr>
        <tr><td>04/01/2015</td></tr>
        <tr><td>05/01/2015</td></tr>
        <tr><td>06/01/2015</td></tr>
        <tr><td>07/01/2015</td></tr>
        <tr><td>08/01/2015</td></tr>
        <tr><td>09/01/2015</td></tr>
        <tr><td>10/01/2015</td></tr>
        <tr><td>11/01/2015</td></tr>
        <tr><td>12/01/2015</td></tr>
    </tbody>
</table>