Your approach seems like you are using a vector-distance based on finding the length of the sum of all independent basis "errors".
The total differential of a multi-variable function describes the infinitesimal change of that function given the infinitesimal changes of all the variables with respect to its derivatives.
What you are doing looks good, but the only thing I would caution you about is what limits you put on the triangle quantities (i.e. the fixed deltas not the infinitesimal values).
Using a small enough value for the triangle delta's of your independent variables should always be OK, but if the function is changing wildly over some region, your errors will be way off. To look at whether this is the case you see how many first derivatives the function has (if any) that are equal to zero and also how the function behaves around your point (whether its derivative suddenly gets really high or really low just after your point).
The method though (calculating the error vector and finding its length) is a sound approach, but the important thing to be aware of is how local this information is and if it's not local (i.e. the stuff that I talked about above happens) then you need to re-consider looking at more global information like later derivatives at that point.
Looking at later derivatives is exactly what is done in numerical analysis when you get the "crazy functions" that have the potential to go wild and if you are interested in error analysis to take into account this behaviour, get some material on numerical analysis and differential equations.