How do you add custom fields defined in posts to the rest API responses in wordpress

Question

I want to do this without using any sort of plugin since these are both core wordpress features (custom fields and the REST API). Here is the documentation for custom fields for reference:

https://codex.wordpress.org/Using_Custom_Fields

Here is a screenshot from my wordpress installation:

Here is what the API response for a post looks like currently:

{
    "_links": {
        "about": [
            {
                "href": "http://example.com/wp-json/wp/v2/types/post"
            }
        ],
        "author": [
            {
                "embeddable": true,
                "href": "http://example.com/wp-json/wp/v2/users/1"
            }
        ],
        "collection": [
            {
                "href": "http://example.com/wp-json/wp/v2/posts"
            }
        ],
        "curies": [
            {
                "href": "https://api.w.org/{rel}",
                "name": "wp",
                "templated": true
            }
        ],
        "replies": [
            {
                "embeddable": true,
                "href": "http://example.com/wp-json/wp/v2/comments?post=21"
            }
        ],
        "self": [
            {
                "href": "http://example.com/wp-json/wp/v2/posts/21"
            }
        ],
        "version-history": [
            {
                "href": "http://example.com/wp-json/wp/v2/posts/21/revisions"
            }
        ],
        "wp:attachment": [
            {
                "href": "http://example.com/wp-json/wp/v2/media?parent=21"
            }
        ],
        "wp:featuredmedia": [
            {
                "embeddable": true,
                "href": "http://example.com/wp-json/wp/v2/media/23"
            }
        ],
        "wp:term": [
            {
                "embeddable": true,
                "href": "http://example.com/wp-json/wp/v2/categories?post=21",
                "taxonomy": "category"
            },
            {
                "embeddable": true,
                "href": "http://example.com/wp-json/wp/v2/tags?post=21",
                "taxonomy": "post_tag"
            }
        ]
    },
    "author": 1,
    "categories": [
        5,
        4
    ],
    "comment_status": "open",
    "content": {
        "protected": false,
        "rendered": ""
    },
    "date": "2017-05-14T15:25:33",
    "date_gmt": "2017-05-14T15:25:33",
    "excerpt": {
        "protected": false,
        "rendered": ""
    },
    "featured_media": 23,
    "format": "standard",
    "guid": {
        "rendered": "http://example.com/?p=21"
    },
    "id": 21,
    "link": "http://example.com/2017/05/14/post/",
    "meta": [],
    "modified": "2017-05-15T18:17:34",
    "modified_gmt": "2017-05-15T18:17:34",
    "ping_status": "open",
    "slug": "",
    "sticky": false,
    "tags": [],
    "template": "",
    "title": {
        "rendered": ""
    },
    "type": "post"
}

In case it could be relevant, here are my active plugins:

Answer 1

I found the REST API Custom Fields plugin useful for this.

Answer 2

First you need to register_rest_fields to adding custom endpoints in WP REST API JSON Response

add_action( 'rest_api_init', 'add_custom_fields' );
function add_custom_fields() {
register_rest_field(
'post', 
'custom_fields', //New Field Name in JSON RESPONSEs
array(
    'get_callback'    => 'get_custom_fields', // custom function name 
    'update_callback' => null,
    'schema'          => null,
     )
);
}

Then define your functions to get custom fields

function get_custom_fields( $object, $field_name, $request ) {
//your code goes here
return $customfieldvalue;
}

Tested on local site