If you want to serve DNS for a subdomain while keeping most of the DNS records to do with its domain with another DNS provider you need NS records.
On the nameserver for example.com:
somesubdomain NS ns1.otherprovider.net somesubdomain NS ns2.otherprovider.net somesubdomain NS ns3.otherprovider.net somesubdomain NS ns4.otherprovider.net
On the otherprovider.net nameserver for somesubdomain.example.com you just need a “normal” A record (+CNAME/MX/whatever):
somesubdomain A 126.96.36.199
I have this setup working with everydns.net providing most of my DNS but linode providing DNS for the sites I have moved off my old providers and on to linode.
The previously wonderful everydns that served me well for many years (for nothing!) has been bought up and is going paid in less than 90 days so I’m taking this opportunity to move things around.