I will begin with some background:

The solutions $\theta$ of $$\cos \theta=x $$ constitute of two families, each of which is an arithmetic progression. Namely, if $\arccos x$ denotes any particular solution then the families are $$\left\{ \arccos x+2 \pi k: k \in \mathbb{Z} \right\} \cup \left\{ -\arccos x+2 \pi k: k \in \mathbb{Z} \right\}. $$ If, instead, we solve the system $$\cos \theta=x \\ \sin \theta=y $$ with $x,y$ satisfying the compatibility condition $$x^2+y^2=1 $$ then the solutions $\theta$ form a single arithmetic progression $$\{\theta^*+2 \pi k: k \in \mathbb{Z} \} $$ where $\theta^*$ is any particular solution. Since $\theta$ is then uniquely defined up to additive integer multiples of $2 \pi$, the differential $\mathrm{d} \theta$ is well-defined, and one can see that $$\mathrm{d} \theta=-y \mathrm{d} x+x \mathrm{d} y. $$

Analogously, one can think of the equation $$\wp(\theta;g_2,g_3)=x $$
as being an underdetermined system, since it has the reflection symmetry $\theta \mapsto -\theta$, as well as the translations $\theta \mapsto \theta+2 \omega_i, \; i=1,2$, where $2\omega_1,2\omega_2$ are generators of the period lattice $$\Lambda=\{2m \omega_1+2n \omega_2:m,n \in \mathbb{Z} \}.$$
If, instead one looks at the system
$$\wp(\theta;g_2,g_3)=x \\ \wp'(\theta;g_2,g_3)=y$$
with $x,y$ satisfying the compatibility condition $$y^2=4x^3-g_2 x-g_3 $$
then $\theta$ is uniquely defined only up to translations by points on the period lattice. In fact, one of the branches of $\theta$ is what Mathematica calls
`InverseWeierstrassP[{x,y},{g2,g3}]`

or $$\wp^{-1} (x,y;g_2,g_3).$$

My questions are:

- Is there a nice formula for $\mathrm{d} \theta$ in this case?
- Similarly, are there simple formulas for the partial derivatives below? $$\frac{\partial}{\partial x} \wp^{-1}(x,y;g_2,4x^3-g_2x-y^2) \\ \frac{\partial}{\partial y} \wp^{-1}(x,y;g_2,4x^3-g_2x-y^2) $$