That's typically only true only if you link to a libmysqlclient version later than 3.23 (maybe later - of the versions I have on my machine right now, 4.0.24 is GPL and 3.23 is LGPL licensed). It's permitted (with a few restrictions) to link any program to an LGPL library. By the way, there seems to be some question as to weather dynamic linking can legally be seen as a derivative work (though I would not try to put my eggs in that basket).
Also, you have no obligations under the GPL or LGPL if you don't distribute your work - both licenses are only activated by distributing a (derivative) work. Merely using MySQL does not force you to do anything.
