When trying to arrive to Euler-Lagrange equation, Susskind does terrific job but I have one problem. Let's consider motion in only $x$ direction with respect to time.
$$x(t) = \hat x(t) + \epsilon f(t)$$ (variation where $\hat x(t)$ is the true trajectory and x(t) the varied one)
$$\frac {dx}{d\epsilon} = f(t)$$
Then we differentiate action with respect to $\epsilon$ to see how action changes when $\epsilon$ changes:
$$\frac{dS}{d\epsilon} = \int \frac{\partial L}{\partial x} \frac{\partial x}{\partial \epsilon}$$ (note that i didn't fully write the second part for $\dot x$ but it should be included.
I wonder now, why did he bring partial derivative for $L$ in $\frac{dS}{d\epsilon}$ but not for $\frac{dx}{d\epsilon}$.
I understand that he does chain rule for $L$ as it's a function of $x$ which is function of $t$, but i never knew that doing chain rule transforms derivative notation from full to partial. would appreciate some help.