  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.
