You receive this quest from the mite gatekeeper in the Old Sycamore cave, where you go when pursuing Tartuccio. Look for her near the exit point in the upper left corner of the map (there are two exits, you need the farthest from the corner). Talk to her, then go the Old Sycamore Depths and look for a giant spider in the upper left corner of the location.
Return to the gatekeeper and give her the mandibles to earn 180 XP and a part of a bridge lift.
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…upper right, not upper left.