A novel residual generator design for fault diagnosis is proposed for linear neutral delay system with one time-varying delay in the state and output, and one constant delay in the state's derivative. Disturbance and fault are represented as structured unknown inputs in the state equation. The resulting residual generator is in the form of a full-order unknown input observer (UIO) that is exponential stable delay-dependently whose existence and stability conditions are presented. A systematic design method is also provided, which is illustrated by a numerical example.