Weak derivative boundary layer ($y_{\mathrm{layer}}=O(\delta ),y_{\mathrm{layer}}^{\prime }=O(1)$)

A correction of amplitude $O(\delta )$ varying on width $O(\delta )$ can be negligible in a function's leading outer approximation while producing an order-one change in its derivative. For example $-\delta e^{-x/\delta }$ has derivative $e^{-x/\delta }$, allowing an endpoint derivative condition to be met without changing the leading outer boundary value. Such weak layers matter in matched asymptotic expansions involving derivative boundary conditions, even when only another endpoint has an order-one layer in the function itself.