In this derivation (and also here) of the Euler-Lagrange equation in the isoperimetric case, one begins with an extremising function $y$ and adds two terms, obtaining $\hat{y} = y + \epsilon_1 \eta_1 + \epsilon_2 \eta_2$, where $\epsilon_i \in \mathbb{R}$ and $\eta_i = \eta_i (x)$. The summand $\epsilon_2 \eta_2$, which is not present in the unconstrained case, is a "correction term", included so that $\hat{y}$ satisfies the isoperimetric constraint, despite the perturbation of $y$ by $\epsilon_1 \eta_1$.
However, the subsequent calculation of $\frac{\partial}{\partial \epsilon_1}$ treats $\epsilon_2$ as being independent of $\epsilon_1$. How can this be? Is the derivation incorrect, or have I misunderstood?