Rational functions that share finite values with their first derivative

We treat shared value problems for rational functions $R(z)$ and their derivative $R^{\mathrm{\prime }}(z)$ in the plane and on the sphere. We also consider shared values for the pair $R(w)$ and ${\mathrm{\partial }}_{z}R=\lambda w\cdot R^{\mathrm{\prime }}(w)$ on $\mathbb{C}\setminus \{0\}$ and $\widehat{\mathbb{C}}$, again with rational functions $R$. In $\mathbb{C}\setminus \{0\}$ this is related to shared values of meromorphic functions $f:\mathbb{C}\to \widehat{\mathbb{C}}$ and $f^{\mathrm{\prime }}$ through $f(z)=R(w)$ with $w=\exp (\lambda z)$, while on $\widehat{\mathbb{C}}$ this is connected to shared limit values in a similar fashion.