Mr. Hyde is driven solely by the sadism of his nature in the murder of Sir Danvers Carew.
It is not until Chapter 10 of The Strange Case of Dr. Jekyll and Mr. Hyde that the reader learns the cause of the death of the well-respected Sir Danvers Carew. The murder has been a random and sadistic attack. This attack is clarified in the transcription of the letter Jekyll leaves for Utterson in the laboratory. This transcription is a full disclosure of his experiment of separating his darker side from his better nature.
In Dr. Jekyll's letter to Utterson, he reveals that he was able to create a compound that would separate his spiritual nature from "the lower elements of [his] soul." This dark alter ego generated from his experiment was Edward Hyde, whose every thought and action was centered upon himself. At first, Jekyll was able to exert control over his darker side; however, he realized that it became more and more difficult to return to himself after each transformation. Then, after he found himself awakening in Hyde's quarters and still possessing Hyde's hairy arms and hands, he decided to stop taking the compound for a while.
After two months, Dr. Jekyll stated that he could no longer resist experimenting again. When he transformed himself after this extended time into Mr. Hyde, his darker side emerged in a wild and vengeful state. It was in this "spirit of Hell" that Mr. Hyde beat Carew mercilessly with his walking stick until he killed the man:
With a transport of glee, I mauled the unresisting body, tasting delight from every blow; and it was not till weariness had begun to succeed, that I was suddenly, in the top fit of my delirium, struck through the heart by a cold thrill of terror. (Ch. 10)
After this incident Dr. Jekyll realized that his darker side had gained dominance over him. So he took a high dose of his compound and composed his full statement.