noun
Refers to the point where something is attached or branches off: the base of a limb, the joint of a finger, the crotch of a tree, etc.
木の付け根にキノコが生えていた。
Mushrooms were growing at the base of the tree.
親指の付け根が痛い。
The base of my thumb hurts.
It broke off at the joint of the branch.
Uses the older 附 character; very rare.
根元 also means 'base' or 'root', but is more often used for the very bottom of something (e.g., a tree trunk, a hair), while 付け根 emphasizes the attachment point or joint.
関節 specifically means 'joint' in an anatomical sense (e.g., knee, elbow). 付け根 is broader and can refer to any base or crotch, not necessarily a movable joint.
Compound of 付け (tsuke, 'attaching') + 根 (ne, 'root'). Literally 'attachment root', referring to the point where something is attached.