I shall assume that in your question, you are asking about *continuous* derivations on a Banach algebra $A$. Results for everywhere-defined-but-not-continuous derivations were studied intensively 30-40 years ago but my understanding is that the remaining open problems are thought to be inaccessible.

So, let $A$ be a Banach algebra and let $D:A\to A$ be a continuous derivation. Without loss of generality, we can assume that $A$ is unital — if not, then we simply adjoin an identity element and extend $D$ by making it $0$ on the adjoined identity.

We then have the following old result of A. M. Sinclair (Proc. AMS 20 (1969), 166–170):

Let $P$ be a primitive ideal of $A$. Then $D(P)\subseteq P$.

(This can also be found as Proposition 2.7.22(ii) of Dales's behemoth *Banach Algebras and Automatic Continuity*, but the original paper of Sinclair is reasonably self-contained and easy to follow.)

**If $A$ is commutative**, then primitive ideals are the same thing as maximal ideals, and so Sinclair's theorem implies that $D(M)\subseteq M$ for every maximal ideal $M$. But since $A={\mathbb C}1_A+M$ and $D(1_A)=0$, we get $D(A)\subseteq M$ for all maximal ideals $M$, which is to say $D(A)\subseteq {\rm rad}(A)$ — and that is the result which you quoted in your question.