Summary: Let \((V,0)=\{(z_1,\ldots,z_n) \in \mathbb{C}^n : f(z_1,\ldots,z_n)=0\}\) be an isolated hypersurface singularity with \(\operatorname{mult}(f)=m\). Let \(J_k(f)\) be the ideal generated by all \(k\)-th order partial derivative of \(f\). For \(1 \leq k\leq m-1\), the new object \(\mathcal{L}_k (V)\) is defined to be the Lie algebra of derivations of the new \(k\)-th local algebra \(M_k(V)\), where \(M_k(V):=\mathcal{O}_n/(f+J_1(f)+\cdots+J_k(f))\). Its dimension is denoted as \(\delta_k (V )\). This number \(\delta_k (V )\) is a new numerical analytic invariant. We compute \(\mathcal{L}_3 (V)\) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of \(\delta_3 (V)\). We also formulate a sharp upper estimate conjecture for the \(\delta_k (V)\) of weighted homogeneous isolated hypersurface singularities and verify this conjecture for large class of singularities. Furthermore, we formulate another inequality conjecture: \( \delta_{(k+1)}(V) < \delta_k(V)\), \(k\geq1\) and verify it for low-dimensional fewnomial singularities.