Let $\Phi(0,\beta)$ a normal function from $On$ to $On$, and let $\Phi(\alpha,\beta)$ be the $\alpha$-th derivative of $\Phi(0,\beta)$. For example, let $\Phi(0,\beta)=\aleph_\beta$. Then, all functions $F_\alpha(\beta)=\Phi(\alpha,\beta)$ are also normal functions for a fixed $\alpha$. I have two questions:
1. Let $G_\beta(\alpha)=\Phi(\alpha,\beta)$. Is this one also a normal function for fixed $\beta$? (Maybe it depends on $\beta$?)
2. Is the diagonalization $\Psi(\alpha)=\Phi(\alpha,\alpha)$ also a normal function?