Answer
See answer below.
Work Step by Step
a) Both are the infinitesimal increments to the independent variable x: $(\partial x)_y=dx$
b) $(\partial z)_y$ is the increment in the dependendant variable caused by a change in y only, while dz refers to the increment caused by a change in both variables. $|(\partial z)_y|\leq|dz|$
c) Reducing the differential relation: $dz=(\partial z)_x+(\partial z)_y$
