"Laws of Area" is vague, but the formula for dtheta is straightforward.
\theta = \arctan (y/x)+C
Here C is an additive constant that depends on 1. Which region of the plane you are in, and 2. How you assign angles to points in the plane. For example, if you define theta to be between 0 and 2pi, then C=0 for (x,y) in the first quadrant, C=pi in the second and third quadrants, and C=2pi in the fourth quadrant. If you define theta to lie between -pi and +pi, then C works differently. But all that is irrelevant, because even though theta as a function of x and y has some ambiguity, no matter what your convention is for assigning angles to points, dtheta is well defined by the formula:
d\theta = \frac{\partial \theta}{\partial x} dx +\frac{\partial \theta}{\partial y}dy
There are ways to derive the formula for dtheta without relying on a formula for theta as a function of x and y. For example, no matter how you assign angles, you have:
x=r\cos(\theta), \hspace{1cm} y=r\sin(\theta).
Use these formulas to calculate dx and dy as linear functions of dr and dtheta (with a basepoint understood to be fixed). You get:
dx = (x/r)dr -yd\theta,\hspace{1cm} dy= (y/r)dr+xd\theta.
Multiply the first equation by y, the second equation by x and subtract to obtain:
ydx - xdy = -(y^2+x^2)d\theta =-r^2d\theta
Dividing out the - (r squared) gives you the formula for dtheta.
