Derivation into dense ideal of Banach algebras

Question

Let $A$ be a Banach algebra and $I$ be an ideal of $A$. A derivation $D\colon A\to I$ is a linear bounded map, with the following property: $$D(ab)=aD(b)+D(a)b,\qquad a,b\in A.$$ Suppose that $I$ is dense in $A$, and any derivation $D\colon A\to I$ is an inner derivation i.e., there is a $x\in I$ such that for all $a\in A$ we have $D(a)=ax-xa.$ Now could we say that any derivation from $A$ into $A$ is an inner derivation? In other word, for derivation $D\colon A\to A$, is there an element $x\in A$, such that $D(a)=ax-xa$, for all $a\in A$?