Hello PF, I was doing the derivation of the Lagrange equation of motion and had to do some calculus of variations. The first step in the derivation is to multiply the integral of ƒ(y(x),y'(x);x)dx from x1 to x2 by δ. and then by the chain rule we proceed. But I cannot understand why we are not applying the chain rule to x too. All the proofs omit this part saying that x1 x2 are fixed and thus it goes to zero. The book I am using is "Classical dynamics of Particles and Systems" by Marion and Thornton. Thank you!