How to set up one or more sub-domain on a different servers

Hello there I need to set up few sub-domains on different servers, what would be the correct way to do this? Main Domain records is handled by BIND locally and domain name is pointing to the server IP using Child Nameservers.

Sub.mysite.com is a server created on another machine with a different IP, I have added an A record as per below however this is not working for me sub.mysite.com. IN A 2.2.2.2

my current BIND Records File looks like this

$ttl 38400
mysite.com. IN  SOA localhost. webmaster.mysite.com. (
            1346411437
            10800
            3600
            604800
            38400 )
@   IN  NS  localhost.
mysite.com. IN  A   1.1.1.1
www.mysite.com. IN  A   1.1.1.1
ftp.mysite.com. IN  A   1.1.1.1
m.mysite.com.   IN  A   1.1.1.1
localhost.mysite.com.   IN  A   127.0.0.1
webmail.mysite.com. IN  A   1.1.1.1
admin.mysite.com.   IN  A   1.1.1.1
mail.mysite.com.    IN  A   1.1.1.1
mysite.com. IN  MX  5 mail.mysite.com.
mysite.com. IN  TXT "v=spf1 a mx a:mysite.com ip4:1.1.1.1 ?all"
sub.mysite.com. IN  A   2.2.2.2 <code>