Orthogonality and smooth points in $C(K)$ and $C_b(\Omega)$

For the usual norm on spaces $C(K)$ and $C_b(\Omega)$ of all continuous functions on a compact Hausdorff space $K$ (all bounded continuous functions on a locally compact Hausdorff space $\Omega$), the following equalities are proved:
$$ \lim_{t\to0+}\frac{\|f+tg\|_{C(K)}-\|f\|_{C(K)}}t=\max_{x\in\{z\mid\,|f(z)|=\|f\|\}}\operatorname{Re}(e^{-i\arg f(x)}g(x)) $$
and
$$ \lim_{t\to0+}\frac{\|f+tg\|_{C_b(\Omega)}-\|f\|_{C_b(\Omega)}}t=\inf_{\delta>0}\sup_{x\in\{z\mid\,|f(z)|\ge\|f\|-\delta\}}\operatorname{Re}(e^{-i\arg f(x)}g(x)). $$
These equalities are used to characterize the orthogonality in the sense of James (Birkhoff) in spaces $C(K)$ and $C_b(\Omega)$ as well as to give necessary and sufficient conditions for a point on the unit sphere to be a smooth point.