Let f(x) = an⁻¹ (( (x2+1))/(sqrt(x2+4))). Find f^(4)(0), the 4th derivative at x=0$.

1. Let $f(x) = \tan^{-1}\left(\frac{\ln(x^2+1)}{\sqrt{x^2+4}}\right)$. Find $f^{(4)}(0)$, the 4th derivative at $x=0$.

Step 1: Analyze the function for symmetry.
Let $g(x) = \frac{\ln(x^2+1)}{\sqrt{x^2+4}}$.
$g(-x) = \frac{\ln((-x)^2+1)}{\sqrt{(-x)^2+4}} = \frac{\ln(x^2+1)}{\sqrt{x^2+4}} = g(x)$.
So