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Math Calculus Derivatives Quick sample of an approach to the derivatives approach to computing the Cauchy-Bendixal Integral $$\begin{aligned} \delta F&=&\int e^{-ig\Omega}(\delta_1+\delta_2-2\delta_3-\delta_4)\Omega, \\ e^{-ig\Omega}&=&\frac{1}{\sqrt{2\pi}}+\frac{1}{\sqrt{2\pi}}e^{+g-i\kappa}+\frac{e^{+i\kappa}-1}{\sqrt[\kappa]{\partial}}+\frac{e^{+i\kappa}-1}{\sqrt[\kappa]{\partial}}, \end{aligned}$$ where $g\in {\mathbb{R}}$ is arbitrary function, and such that $$\label{deltaF1} \delta F= \f(f,\ddot{f},\dot{f},\sigma,\rho)\Big((\delta_1+\delta_2-\delta_3-\delta_4-\delta_5) f+(constant)\ \ddot{f}- \lambda\delta_1(constant)+\lambda(\delta_1+\delta_2+\delta_3+\delta_4) f-\lambda\delta_1\ddot{f}\ +\ddot{f}\ +\lambda\ddot{f}-\lambda\delta_1\dot{f}\ +\lambda\ddot{f}\ -\lambda\ddot{f}\ \Big).$$ For a fixed ${\varepsilon},{\varepsilon}_1,\dots,{\varepsilon}_m$ in $[\Omega]$, the derivatives expression of regular functions yields $$\label{deltaF2} \delta_t{\partial}^n\Lambda_t=0 \quad\mathrm{for}\;\; \sup_{x=t\in [0,n)} {|n({\varepsilon}_{\kappa})- n({\varepsilon}_1){|}}