Die Kinder lächelten, was deren Dankbarkeit zeigte.

Questions & Answers about Die Kinder lächelten, was deren Dankbarkeit zeigte.

Why use was instead of das here?
The word was is used as a relative pronoun when it refers back to an entire clause or idea rather than a specific noun. Here, it refers to the whole preceding event—the fact that the children smiled. If it referred to a specific neuter noun, you would use das.
Could I just say ihre Dankbarkeit instead of deren Dankbarkeit?
Yes, absolutely! The standard possessive article ihre (their) is completely fine here. Using the demonstrative pronoun deren (literally 'of those') simply places a bit more emphasis on the children. In sentences with multiple nouns, deren is often used to prevent ambiguity about who owns what, though in this simple sentence it just sounds slightly more emphatic.
Since the lesson is about derer vs. deren, why must we use deren in this sentence?
The key rule is direction: deren points backward to something already mentioned, while derer points forward to a further explanation (usually a relative clause). Because we are referring back to the children we just talked about, we must use deren. You would use derer if you were saying 'the gratitude of those, who...' (die Dankbarkeit derer, die...).
Why is the verb zeigte at the very end of the sentence?
The word was introduces a relative clause. Relative clauses are subordinate clauses, which means the conjugated verb (zeigte) must be placed at the very end of the clause.

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