Added sympy. Fixed derivation issue.

content/method/method.tex

@@ -602,12 +602,10 @@ Therefore, our derivative to calculate from before is equal to:
   \frac{\partial \hat{F}_0^{-1}(\alpha ; \mathcal{B}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{1}{\sigma^{-1}'\left( \sigma(\tau - f_{\theta}(\mathbf{x}_i)) \right)\cdot\left( \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma'(\tau - f_{\theta}(\mathbf{x}_j))\right) + 1 }
 \]
 
-Recalling the inverse derivative identity $\frac{1}{(f^{-1})'(f(x))} = f'(x)$, we note that $\frac{1}{\sigma^{-1}'(\sigma(\tau - f_{\theta}(\mathbf{x}_i)))} = \sigma'(\tau - f_{\theta}(\mathbf{x}_i))$.
-
-Therefore:
+Substituting $\sigma^{-1}'(\sigma(u)) = \frac{1}{\sigma'(u)}$ and multiplying numerator and denominator by $\sigma'(\tau - f_{\theta}(\mathbf{x}_i))$:
 
 \[
-  \frac{\partial \hat{F}_0^{-1}(\alpha ; \mathcal{B}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{\sigma'(\tau - f_{\theta}(\mathbf{x}_i))}{\left( \sum_{\mathbf{x}_j \in \mathcal{B}_0 ; j \neq i}\sigma'(\tau - f_{\theta}(\mathbf{x}_j))\right) + 1}
+  \frac{\partial \hat{F}_0^{-1}(\alpha ; \mathcal{B}_0)}{\partial f_\theta(\mathbf{x}_i)} = \frac{\sigma'(\tau - f_{\theta}(\mathbf{x}_i))}{\displaystyle\sum_{\mathbf{x}_j \in \mathcal{B}_0}\sigma'(\tau - f_{\theta}(\mathbf{x}_j))}
 \]
 
 Which we can plug back into the ROLL derivation framework.

flake.nix

@@ -58,7 +58,12 @@
     in
     {
       devShells.${system}.default = pkgs.mkShell {
-        buildInputs = [ (pkgs.texlive.combine texPkgs) pkgs.culmus pkgs.fontconfig ];
+        buildInputs = [
+          (pkgs.texlive.combine texPkgs)
+          pkgs.culmus
+          pkgs.fontconfig
+          (pkgs.python3.withPackages (ps: [ ps.sympy ]))
+        ];
         shellHook = ''
           export FONTCONFIG_FILE="${fontsConf}"
         '';