This may be physics related but I think it belongs here because I have some doubt about mathematical operators we have gauss law in differential form as $$\nabla \cdot E = \frac{\rho } {\epsilon_0}$$ now I want to convert it in integral form so I wrote it as $$\frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y}+ \frac{\partial E_z}{\partial z} = \frac{\rho}{\epsilon_0 }$$ Now I multiplied it with $dxdydz$ So $$\frac{\partial E_x . dxdydz}{\partial x} + \frac{\partial E_y.dxdydz}{\partial y}+ \frac{\partial E_z.dxdydz}{\partial z} = \frac{\rho dxdydz}{\epsilon_0 }$$ Which becomes $$\int{\partial E_x .dydz+ \partial E_y.dxdz+ \partial E_z.dxdy} = \frac{Q}{\epsilon_0 }$$
Which takes form of $$\int \vec{E}.\vec{ds}= \frac{Q}{\epsilon_0}$$
So was my process correct? Does $$\frac{ dxdydz}{\partial x}= dydz ?$$
You are applying the divergence theorem. Your approach looks fine, but I would not say that $$ \frac{dx\, dy\, dz}{\partial x} = dy \, dz, $$ since it is so meaningless that it would deserve a whole theory to make it rigorous. Since $\nabla \cdot E$ is the divergence of $E$, and since $Q=\int \rho\, dx\, dy\, dz$, just apply the divergence theorem.