Using username and password authentication

dynamic_dnsimple.sh

#!/bin/bash
set -e

domain=mislav.net
record=511770     # mysubdomain.mislav.net

ip="$( curl -s icanhazip.com )"
payload="{ \"record\": { \"content\": \"${ip}\" } }"

curl -fsS --netrc \
     -H 'Accept: application/json' \
     -H 'Content-Type: application/json' \
     -X PUT -d "${payload}" \
     "https://dnsimple.com/domains/${domain}/records/${record}"

netrc

# part of ~/.netrc
machine dnsimple.com
  login mislav.marohnic@gmail.com
  password YOUWISH