Generative recommendation provides a novel paradigm in which each item is represented by a discrete semantic ID (SID) learned from rich content. Most methods treat SIDs as predefined and train recommenders under static indexing. In practice, SIDs are optimized only for content reconstruction rather than recommendation accuracy. This leads to an objective mismatch : the system optimizes an indexing loss to learn the SID, and a recommendation loss for interaction prediction, but because the tokenizer is trained independently, the recommendation loss cannot update it. A natural approach is to make semantic indexing differentiable so recommendation gradients can directly influence SID learning, but this often causes codebook collapse with only a few codes used. We attribute this to early deterministic assignments that limit codebook exploration, leading to imbalance and unstable optimization. In this paper, we therefore propose DIGER (Differentiable Semantic ID for GEnerative Recommendation). DIGER is a first step towards an effective differentiable semantic ID for generative recommendation. The Gumbel noise explicitly encourages early-stage exploration over codes, mitigating collapse and improving code utilization. To better balance exploration and convergence, we introduce two uncertainty decay strategies that reduce the Gumbel noise, enabling a gradual shift from early-stage exploration to the exploitation of learned SIDs. Extensive experiments across multiple public datasets demonstrate consistent improvements from differentiable semantic ID. These results confirm the effectiveness of aligning indexing and recommendation objectives through differentiable SIDs. This identifies differentiable SID as a promising area of study. Our code is released under https://github.com/junchen-fu/DIGER.