f = f(x, y)

df = f(x + dx, y + dy) - f(x, y)
= f(x + dx, y + dy) - f(x, y + dy) + f(x, y + dy)- f(x, y)
= f_x.dx + f_y.dy

because evaluating the partial derivative at y+dy instead of y doesn't make any difference, right?