Learn Danish: Lærerens taske er derinde. - The teacher's bag is in there.

It is a compound of der (there) and inde (inside). Danish frequently combines her (here) or der (there) with directional words to create highly specific location markers, so derinde translates perfectly to "in there".

Why does derinde end in an -e? Could I just say derind?

You must use derinde here because the bag's position is static; it is just sitting there. In Danish, spatial adverbs ending in -e (like inde, ude, oppe) indicate location or being at rest. The forms without the -e (like ind, ud, op) indicate movement toward a destination. Since the verb is er (is), there is no movement, so we need the -e.

Why is it taske instead of the definite form tasken, since we are talking about a specific bag?

Because it comes right after a possessive word (lærerens). In Danish, whenever a noun is preceded by a possessive (like my, your, or an -s ending), the noun itself must drop its definite suffix and stay in the indefinite form. The possessive word already makes it definite in meaning.

How exactly is the word lærerens constructed?