Solve $\frac{d}{dx}f(x)=f(x-1)$ I am trying to find a function such that 
$\dfrac{d}{dx}f(x)=f(x-1)$ 
Is there such function other than $0$ ?

Certainly. Let $b$ be the unique real number such that $b=e^{-b}.$ Then for any real $a,$ the function $$f(x)=ae^{bx}$$ satisfies the desired property. In fact, for real-valued functions on the reals, only functions of this form will satisfy the desired property. (As achillehui points out in the comments, there are other alternatives if we consider complex-valued functions.)
In particular, $b=W(1)$, where $W$ is the Lambert W function. $W(1)$ is also sometimes known as the Omega constant.
