Gateaux derivative of C* norm

We find an expression for Gateaux derivative of the $C^*$-algebra norm. This gives us alternative proofs or generalizations of various known results on the closely related notions of subdifferential sets, smooth points and Birkhoff-James orthogonality for spaces $\mathscr B(\mathcal H)$ and $C_b(\Omega)$. We also obtain an expression for subdifferential sets of the norm function at $A\in\mathscr B(\mathcal H)$ and a characterization of orthogonality of an operator $A\in\mathscr B(\mathcal H, \mathcal K)$ to a subspace, under the condition $dist(A, \mathscr K(\mathcal H))< \|A\|$ and $dist(A, \mathscr K(\mathcal H, \mathcal K))< \|A\|$ respectively.