Let $A$ and $B$ be positive operators and $A\ \natural_r\ B =A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^rA^{\frac{1}{2}}$ is a path going through $A$ and $B$. The tangent of $A\ \natural_r\ B$ at $r$ is given by $S_r(A|B)=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^r(\log A^{-\frac{1}{2}}BA^{-\frac{1}{2}})A^{\frac{1}{2}}$ and especially the case $r=0$ is the relative operator entropy. We can find the behavior of $S_r(A|B)$, for $r \in [n,\ n+1]$, is similar to the case $r \in [0,\ 1]$. So we can extend several relations known for $r \in [0,\ 1]$ to $r \in [n,\ n+1]$.