Some background for this question. Given a Riemannian manifold $(M,g)$ and a compact Riemannian manifold $(N,h)$ together with an isometric embedding $(N,h) \hookrightarrow (\mathbb{R}^N, g_{\mathrm{euc}})$, one can define the Sobolev spaces of mappings between manifolds as
$W^{k,p}(M,N) = \{ u \in W^{k,p}(M,\mathbb{R}^n) \,\, | \,\, u \in N \, \mathrm{a.e} \}.$
This definition depends on the embedding of $N$, but one can show that different embeddings result in equivalent definitions in an appropriate sense.
Now, I'm trying to understand whether the function $u(z) = e^{1/z} : B(0,1) \rightarrow \mathbb{CP}^1$ is in $W^{1,p}(B(0,1), \mathbb{CP}^1)$ for some $1 \leq p < 2$, where we put the Fubini-Study metric on $\mathbb{CP}^1$ and use the Euclidean metric on the unit disk $B(0,1) \subset \mathbb{C}$. Writing explicitly the embedding of $\mathbb{CP}^1$ into $\mathbb{R}^3$ and choosing one of the coordinates, the question is translated, up to some constants, into whether the following function
$f(x,y) = \frac{2e^{\frac{x}{x^2+y^2}} \cos \left( \frac{y}{x^2+y^2} \right)}{1+e^{\frac{2x}{x^2+y^2}}} = \mathrm{sech}\left(\frac{x}{x^2+y^2}\right)\cos\left(\frac{y}{x^2+y^2}\right)$
belongs to $W^{1,p}(B)$. I'm somewhat lost with the calculations here. Can this be answered without much technical work?