TITLE: How to take partial derivatives of functions whose inputs depend on the same variable?
QUESTION [7 upvotes]: I am starting to learn about the Calculus of Variations and the Euler-Lagrange equation is extremely confusing to me:

The Euler–Lagrange equation, then, is given by
  $$L_x(t,q(t),q'(t))-\frac{\mathrm{d}}{\mathrm{d}t}L_v(t,q(t),q'(t)) = 0.$$
  where $L_x$ and $L_v$ denote the partial derivatives of $L$ with respect to the second and third arguments, respectively.

Specifically, I don't understand the meaning of:
$$L_x = \frac{\partial L(t, q, q')}{\partial q}$$
I'm confused because $q$ and $q'$ are both functions of $t$, but taking a partial derivative requires that we hold other parameters constant.
Yet $t$ is a parameter of $L$ as well. Thus when differentiating $L$ with respect to $q$, we hold $t$ constant.
But if $t$ is constant then doesn't that mean $q$ and $q'$ are constant?
And hence mustn't  the derivatives $L_x$ and $L_v$ both be equal to zero?

REPLY [5 votes]: Perhaps better for CS backgrounds:  $\partial/\partial t \ F(t,x,y)$ means the derivative of the function with respect to variation in its first slot, the formal parameter $t$.  $\mathrm{d}/\mathrm{d} t\ F(t,x,y)$ means the derivative of the function with respect to the system parameter $t$.  The difference is between local parameter names (partial differentiation) and global variables (total differentiation).