Gateaux derivative of $B(H)$ norm

We prove that for Hilbert space operators $X$ and $Y$, it follows that

where $H_\varepsilon=E_{X^*X}((\vert\vert X\vert\vert-\varepsilon)^2,\vert\vert X\vert\vert^2)$. Using the concept of $\varphi$-Gateaux derivative, we apply this result to characterize orthogonality in the sense of James in $B(H)$, and to give an easy proof of the characterization of smooth points in $B(H)$.