A note on ordinal exponentiation and derivatives of normal functions

Michael Rathjen and the present author have shown that $\Pi^1_1$-bar induction is equivalent to (a suitable formalization of) the statement that every normal function has a derivative, provably in $\mathbf{ACA_0}$. In this note we show that the base theory can be weakened to $\mathbf{RCA_0}$. Our argument makes crucial use of a normal function $f$ with $f(\alpha)\leq 1+\alpha^2$ and $f'(\alpha)=\omega^{\omega^\alpha}$. We will also exhibit a normal function $g$ with $g(\alpha)\leq 1+\alpha\cdot 2$ and $g'(\alpha)=\omega^{1+\alpha}$.