variation in the action question

nobraner

In the Nambu-Goto action, the Lagrange Density is L(dX/dt,dX/dx). When the action is varied, this term becomes a sum of conjugate momentums multiplied by their respective instantaneous velocities, i.e. (Px*dX/dx + Pt*dX/dt). Nowhere, can I find an detailed explanation of how one gets from the first step to the second step. Several references mentioned a Taylor expansion, but otherwise, it seems to be assumed that this step is obvious.

Could anyone direct me to material giving an explicit, step-by-step derivation.