If I have 2 different systems of coordinates: polar and cartesian.

And I want to linearize a function f(x,y) in cartesian coordinate:

f(x,y)= (df/dx)x + (df/dy)y (I)

And in polar coordinate:

f(p,o)= (df/dp)p + (df/do)o (II)

The linearization in (I) is equal to (II)? if is equal, there is a way to proof (II) from (I)?