HMMT 2010 K5

Let the functions $f(\alpha, x)$ and $g(\alpha)$ be defined as

$$f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2}$$

Then $g(\alpha)$ is a polynomial in $\alpha$. Find the leading coefficient of $g(\alpha)$.