The cost of a database server depends on the server requirements of the database. Minimum server requirements depend on the size of the database and the workload being exerted upon the database. MySQL can run with as little as 2GB of disk space and 128MB of RAM.
There is a calculator at http://www.mysqlcalculator.com/ which will help you determine a rough estimate of the amount of RAM needed for your database server.
Zumiv’s Cloud VDS with 8GB of RAM and 110GB SSD start at just $29 per month and would provide sufficient resources for a modest database server setup.
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