You should check the derivation of the Euler Equation line by line. We first introduce , where is a function of that vanishes at the boundary. Next, we form the differential quotient, , we then differentiate with respect to . Now our , , are all fixed NUMBERS! is only to denote that the function is of the form , and we are taking the partial derivative . Actually, some books just use this notation instead.

To give yourself a better understanding, consider differentiating with respect to .