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Questions & Answers about Wir danken der Frau.
Why does it say der Frau? Isn't Frau feminine, so it should be die Frau?
Frau is indeed feminine! However, this sentence is in the dative case. In the dative case, the feminine article die changes to der. It can look confusing because der is normally the masculine article for subjects, but here it is just the dative form of die.
Why is the dative case used here? Doesn't the woman receive the action?
In German, certain verbs automatically require the dative case for their object, and danken (to thank) is a very common one. Instead of thinking 'We thank the woman', it helps to imagine it translates literally as 'We give thanks to the woman'.
If I wanted to say 'We thank the man', would it also use der?
No, because Mann is masculine. The masculine article der changes to dem in the dative case. So, 'We thank the man' would be Wir danken dem Mann.
How would this sentence change if we were thanking multiple women?
It would become Wir danken den Frauen. In the plural dative case, the article changes to den. Also, plural nouns usually get an extra -n added to the end in the dative case (though Frauen already ends in -n in its normal plural form, so it doesn't change further).