Let $X$ be a smooth projective variety over a field $K$. Let $K_0(X)$ denote the Grothendieck group of the abelian category of coherent sheaves on $X$. Let $\newcommand{\Pf}{\mathbf{Pf}}K_0(\Pf(X))$ denote the Grothendieck group of the triangulated category of perfect complexes of sheaves on $X$, or equivalently the bounded derived category of coherent sheaves.
It is well-known that $K_0(X)$ and $K_0(\Pf(X))$ are isomorphic. I would like to cite a reference for this fact, but I can't find one. Does anyone know a place where this is proved?