I was trying to derive Euler Lagrange Form which looks like
$$\frac{\partial L}{\partial q}=\frac{d}{dt}\frac{\partial L}{\partial \dot{q}}$$
I was using a function which I got from here,
$$S(q)=\int_{t_0}^{t_1} L(q,\dot{q},t)dt$$
I was using Feynman technique since I know that rather than simple integration.
$$\frac{dS}{dt}=\int_{t_0}^{t_1} (\frac{\partial L}{\partial q}\dot{q}+\frac{\partial L}{\partial \dot{q}}\frac{\partial }{\partial t}\dot{q} )dt$$
Total time derivative of that function is $0$ so, writing $\frac{dS}{dt}=0$ $$0=\dot{q}(\frac{\partial L}{\partial q}+\frac{\partial}{\partial t}\frac{\partial L}{\partial \dot{q}})$$
$$ \frac{\partial L}{\partial q}+\frac{\partial }{\partial t}\frac{\partial L}{\partial \dot{q}}=0$$
I believe there's difference between total time derivative and short ($\frac{\partial}{\partial t}$) time derivative. Without caring of that I found that my equation is closely related to Euler Lagrange. We have only difference between sign.