For a Boolean function $f:\{0,1\}^n\to\{0,1\}$, the higher-order Boolean derivative $D_S f$ computes the parity of $f$ over each $S$-dimensional subcube. We prove that $D_S f\equiv 1$ exactly when $S$ is a maximal monomial support in the algebraic normal form of $f$. This correspondence motivates the derivative certificate depth $\Delta_\partial(f)$, defined as the minimum degree of a nonempty maximal algebraic normal form monomial. We study the decision problem BCD: given $f$ and $k$, decide whether $\Delta_\partial(f)\le k$. We show that this problem is coNP-complete for every fixed constant $k\ge 1$, NP-complete when $k=n$, and belongs to $\Sigma_2^{\oplus P}$ when $k$ is part of the input; in the variable-$k$ case it is also both NP-hard and coNP-hard. These results identify the $\exists\forall\oplus$ upper bound and leave $\Sigma_2^{\oplus P}$-completeness as an open problem.