Co-exploration of an optimal neural architecture and its hardware accelerator is an approach of rising interest which addresses the computational cost problem, especially in low-profile systems. The large co-exploration space is often handled by adopting the idea of differentiable neural architecture search. However, despite the superior search efficiency of the differentiable co-exploration, it faces a critical challenge of not being able to systematically satisfy hard constraints such as frame rate. To handle the hard constraint problem of differentiable co-exploration, we propose HDX, which searches for hard-constrained solutions without compromising the global design objectives. By manipulating the gradients in the interest of the given hard constraint, high-quality solutions satisfying the constraint can be obtained.