Step-by-Step Solution
Key Concept: For differentiability at $x=0$: continuity requires $\tan(a^4/(b+1))=b+1$ and the derivatives must match. The condition simplifies to: $\sin(b+1) = -\cos(b+1)$ (i.e., $\tan(b+1)=-1$) AND $\cos(a^4)=\pm 1/\sqrt{2}$, giving $a^4=\pi/4+k\pi/2$.
$a=\pm\sqrt{2}$, each giving 5 valid values of $b\in[0,10\pi]$. Total: $2\times 5=\mathbf{10}$ ordered pairs.
Correct Answer: 10