Let $A$  be  a  unital $C^*$ algebra. Assume that $D:A\to A$ is a bounded derivation.

Can one say that $1$ can not be in the image of $D$?

If the answer is no:

 What is a counter example? What kind of $C^*$ algebra admits  outer bounded derivation  but stil they satisfy the above prevent property? 

**Motivation:** I had intention to consider a simillar process to generat a kind of Legendre polynomial(a similar but not identical to it). So I wondered if requirement  $D(P_1)=P_0$ is feasible for some appropriate derivation. On the other hand the impossibility $[x,y]=1$ leads us to  search for outer derivation