Existing work on search result diversification typically falls into the "next document" paradigm, that is, selecting the next document based on the ones already chosen. A sequential process of selecting documents one-by-one is naturally modeled in learning-based approaches. However, such a process makes the learning difficult because there are an exponential number of ranking lists to consider. Sampling is usually used to reduce the computational complexity but this makes the learning less effective. In this paper, we propose a soft version of the "next document" paradigm in which we associate each document with an approximate rank, and thus the subtopics covered prior to a document can also be estimated. We show that we can derive differentiable diversification-aware losses, which are smooth approximation of diversity metrics like alpha-NDCG, based on these estimates. We further propose to optimize the losses in the learning-to-rank setting using neural distributed representations of queries and documents. Experiments are conducted on the public benchmark TREC datasets. By comparing with an extensive list of baseline methods, we show that our Diversification-Aware LEarning-TO-Rank (DALETOR) approaches outperform them by a large margin, while being much simpler during learning and inference.