  We study the norm derivatives in the context of Birkhoff-James orthogonality
in real Banach spaces. As an application of this, we obtain a complete
characterization of the left-symmetric points and the right-symmetric points in
a real Banach space in terms of the norm derivatives. We obtain a complete
characterization of strong Birkhoff-James orthogonality in $\ell_1^n$ and
$\ell_\infty^n$ spaces. We also obtain a complete characterization of the
orthogonality relation defined by the norm derivatives in terms of some newly
introduced variation of Birkhoff-James orthogonality. We further study
Birkhoff-James orthogonality, approximate Birkhoff-James orthogonality,
smoothness and norm attainment of bounded bilinear operators between Banach
spaces.

Functional Analysis