Use get_the_category function , witch will return all categories asigned to a post ( this means all parents and childs allso ) so you can loop thru them and see witch one is parent and witch one is child and print the one you're trying to get . I suggest you build a function in you're theme functions file.
Update
For example let's say you whant to display the child category name in you're single.php theme file so you would do this:
<?php $child_category = post_child_category(get_the_ID()); ?>
<?php if ( $child_category ) echo $child_category->cat_name; ?>
In order for that to work you need to define post_child_category function in you're theme functions file ( if you look in you're theme directory you'll see a functions.php file , if not then you can create it now ) , so you would add the following :
if ( ! function_exists( 'post_child_category' ) )
{
function post_child_category( $id = null )
{
if ( $id = null )
return false;
$categories = get_the_category( $id );
if ( count($categories) > 0 )
{
return $categories[count($categories)-1];
} else {
return false;
}
}
}
Update
If you whant to display the category link you would do this :
<?php $child_category = post_child_category(get_the_ID()); ?>
<?php if ( $child_category ) : ?>
<a href="<?php echo get_category_link($child_category->cat_ID); ?>" title="<?php echo $child_category->cat_name;?>">
<?php echo $child_category->cat_name;?>
</a>
<?php endif;?>