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. 

The maid across the street bears witness to this horrific scene, and she watches helplessly as Hyde "all of a sudden... broke out in a great flame of anger, stamping with his foot, brandishing the cane, and carrying on ... like a madman" (Chapter 4). 

Mr. Hyde murders Sir Danvers Carew in an evil fit of rage, beating the old, white haired man to death with a cane. 

"And next moment, with ape-like fury, he was trampling his victim under foot, and hailing down a storm of blows, under which the bones were audibly shattered and the body jumped upon the roadway" (Chapter 4).

This moment in the novel truly reveals the depravity of Mr. Hyde, that he murders an innocent man without cause, solely to prove his dominance and power over good.  The diction of the murder scene, using words and phrases such as "ape-like fury" and "trampling," reveals a ferocious, animalistic connotation, as if Hyde has lost his humanity.  Hyde's evil nature takes absolute control in this scene, and he acts out with complete abandon.