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It is possible that you might have developed inflammation of the soft tissues of the hand or wrist. Some structures that may be involved include the tendons, ligaments, bursas (fluid filled structures which act as cushions), etc.
The best way to know the cause would be to consult your doctor for a examination, and to confirm the exact condition.
In the mean time, some options for self treatment may include: Splinting of the hand (to help prevent wrist movement and hence the pain) ; Oral anti-inflammatory medications (to reduce the swelling and pain); Changing position of a computer keyboard, or other ergonomic changes and if the symptoms are very severe avoiding working on the laptop for a few days. Application of Ice will also be very helpful.