Given a function :
we want to find the points of the domain of where the value of is smaller (for minima, or larger for maxima) than all other points in some neighbourhood of .
I think the symbol is usually just read as "D" as in "d f d x" for .
This notation is not so common in basic mathematics, but it is so incredibly convenient, especially with Einstein notation as shown at Section "Einstein notation for partial derivatives":
This notation is similar to partial label partial derivative notation, but it uses indices instead of labels such as , , etc.
The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.