This paper introduces a novel approach for efficiently distilling LLMs into smaller, application-specific models, significantly reducing operational costs and manual labor. Addressing the challenge of deploying computationally intensive LLMs in specific applications or edge devices, this technique utilizes LLMs' reasoning capabilities to generate labels and natural language rationales for unlabeled data. Our approach enhances both finetuning and distillation by employing a multi-task training framework where student models mimic these rationales alongside teacher predictions. Key contributions include the employment of zero-shot prompting to elicit teacher model rationales, reducing the necessity for handcrafted few-shot examples and lowering the overall token count required, which directly translates to cost savings given the pay-per-token billing model of major tech companies' LLM APIs. Additionally, the paper investigates the impact of explanation properties on distillation efficiency, demonstrating that minimal performance loss occurs even when rationale augmentation is not applied across the entire dataset, facilitating further reductions of tokens. This research marks a step toward the efficient training of task-specific models with minimal human intervention, offering substantial cost-savings while maintaining, or even enhancing, performance.