How to show button on div mouse hover

Use the below selector

button {
  display: none; /* Hide button */
}

.divbutton:hover button {
   display: block; /* On :hover of div show button */
}

Demo

Also make sure you assign some height or min-height to your div element, else it will be 0 as it doesn't hold any content. Also, don't use display: none; as inline style, as inline styles have highest specificity, and hence, you will be forced to use !important which is bad.

In the above example am using button {/*Styles*/} but that is a general element selector, so make sure you define a class to your button element.

Use following jQuery to perform your task.

Here is a jsfiddle demo

$(document).ready(function () {
    $(document).on('mouseenter', '.divbutton', function () {
        $(this).find(":button").show();
    }).on('mouseleave', '.divbutton', function () {
        $(this).find(":button").hide();
    });
});

Mr. Alien's answer gives a nice CSS implementation. If you need in jquery, use this -

$( ".divbutton" )
 .on("mouseenter", function() {
  $("button").show();
})
.on("mouseleave", function() {
  $("button").hide();
});

In pure JavaScript -

var buttonDiv = document.getElementsByClassName("divbutton")[0];  //better use some id and then use getElementById

buttonDiv.onmouseover = function() {
    document.getElementById("YourButtonId").style.display = 'block';
}

buttonDiv.onmouseout = function() {
    document.getElementById("YourButtonId").style.display = 'none';
}