Yes, this is the right place. Edinburgh is in Midlothian.
However finding the burial place may not be quite so easy.
The burial records are in the care of Edinburgh Council but without knowing which burial ground it is impossible to search. You have to tell them which cemetery you want to took in, and some of the books are so big and heavy that they have special trolleys for bringing them out to you!
If there is a gravestone, you might find that someone has photographed and/or transcribed it. However not everyone has a stone.
I would have a look in
http://www.gravestonephotos.com/public/location.php?area=Lothian&country=Sc&location=Edinburgh&scrwidth=1200http://www.findagraveinscotland.com/homehttp://www.findagrave.com/http://www.memento-mori.co.uk/If there isn't a stone it will be harder.
There might perhaps have been a notice in
The Scotsman newspaper, which can be searched online. There is a charge, but some RootsChatters have subscriptions to it and would be able to tell you whether or not there is such a notice if you tell us the name and exact date of death.
If that fails, then you will need to find out which burial grounds were near where your relative lived. Or, if they came into the city from outside, might they have been taken home for burial?
