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You are on the right track with your post... Compute two slopes then somehow use those slopes to find a normal. The slopes are the result of finite difference (like central difference) But why two slopes? Because we can use them to construct two independent vectors whose cross product will become our normal vector. One vector is in the texture's $u$ ...
I have rewritten my answer to make it easier to follow and I also fixed some mistakes. Let $S:[0,1]^2 \rightarrow \mathbb{R}^3$ be some parameterization of your surface (assume that the derivatives do not become zero anywhere, i.e. it's regular/an immersion). That is, the set of points of your surface is \$\mathcal{M} = \{S(\beta, \gamma) \,:\, (\beta, \gamma)...