What is the actual meaning of $\frac{\partial}{\partial{x}}$ When I searched total derivative on wikipedia it says:
For $L=L(t,x_1(t),x_2(t),x_3(t)...x_n(t))$ the total derivative is given by:
$$\dfrac{\rm{d}L}{\rm{d}t}=\dfrac{\partial L}{\partial t}+\sum_i^n\dfrac{\partial L}{\partial x_i}\dfrac{\partial x_i}{\partial t}~.$$
So here $\dfrac{\partial L}{\partial t}$ is derivative of $L$ w.r.t an explicit independent variable $t$.
However, when I look for generalized chain rule, wikipedia says:
For $$y=y\left(u_1(x_1,x_2,...,x_i),u_2(x_1,x_2,...,x_i),...,u_m(x_1,x_2,...,x_i)\right)$$
the chain rule is given by: $$\frac{\partial y}{\partial x_i}=\sum_{l=1}^m\dfrac{\partial y}{\partial u_l}\dfrac{\partial u_l}{\partial 
x_i}$$
Now the partial derivative operator $\dfrac{\partial}{\partial{x_i}}$ looks more like a total derivative, rather than a derivative w.r.t an explicit independent variable. 
Could anybody tell me what on earth is $\dfrac{\partial}{\partial{x_i}}$ ? 
Thanks!
 A: In the second expression $\frac{\partial y}{\partial x_i}$ is in fact a partial derivative too, and this is because $y$ is ultimately a function of many variables $x_1,\ldots,x_n$.  In your first expression $\frac{dL}{dt}$ a total derivative is possible here because the function is ultimately a function of a single variable, namely $t$.  This is the main reason why a total derivative for $y$ wouldn’t be possible in this case: $L$ ultimately is a function of a single variable while $y$ is a multivariable function.
A: The crucial difference between partial and total derivatives is that for total derivatives there must only be one "independent variable" such as $t$, whereas for partial derivatives we vary one such variable while holding others constant. If we hold the $x_i$ constant as $t$ varies, $L$'s rate of change with respect to $t$ would be $\frac{\partial L}{\partial t}$; if we let the $x_i$ vary as specific functions of $t$, this has a knock-on effect on $L$'s rate of change with respect to $t$, giving it the in general different value $\frac{dL}{dt}$.
The generalized chain rule you found is in a context where no variable is uniquely privileged in the way $t$ is, but we can still relate various partial derivatives. What's more, not only are the two equations you've compared consistent, they do something wonderful when put together. But to do that, we'll need to consider the case $m=n$.
If we have two $n$-dimensional systems $x_i,\,u_l$ of $t$-dependent coordinates with $n\ge2$, we can use$$\frac{\partial y}{\partial x_i}=\sum_l\frac{\partial y}{\partial u_l}\frac{\partial u_l}{\partial x_i}$$(with $m=n,\,y=L$) in$$\frac{dL}{dt}=\frac{\partial L}{\partial t}+\sum_i\frac{\partial L}{\partial x_i}\frac{\partial x_i}{\partial t}$$to obtain an equation analogous to the latter,$$\frac{dL}{dt}=\frac{\partial L}{\partial t}+\sum_{il}\frac{\partial L}{\partial u_l}\frac{\partial u_l}{\partial x_i}\frac{\partial x_i}{\partial t}=\frac{\partial L}{\partial t}+\sum_l\frac{\partial L}{\partial u_l}\frac{\partial u_l}{\partial t},$$as per $\frac{\partial u_l}{\partial t}=\sum_i\frac{\partial u_l}{\partial x_i}\frac{\partial x_i}{\partial t}$, the $L=u_l$ special case with $\frac{du_l}{dt}=0$. So the Liouvillian $\frac{dL}{dt}-\frac{\partial L}{\partial t}$ of $L$ looks very similar regardless of the coordinate system; you can write it as either $\sum_i\frac{\partial L}{\partial x_i}\frac{\partial x_i}{\partial t}$ or $\sum_l\frac{\partial L}{\partial u_l}\frac{\partial u_l}{\partial t}$.
