Taylor Expansion of the 1/2th Derivative

Question

In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? The Taylor series for the $1/2th$ derivative seems to be: $\sum\limits_{n=0} {1/2 \choose n} (D-1)^n $ with $D$ being $\frac{d}{dx}$. The one is there because if you center the Taylor series around zero it will acquire a lot of sums of infinities. Because $D$ is really a matrix, and therefore $\sqrt D$ is also a matrix, so I think that 1 should really be the identity matrix. Any help would be appreciated.