Let $\mathbb{R}$ denote the set of all real numbers. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(0,4)$ be functions defined by
$ f(x)=\log _e\left(x^2+2 x+4\right), \text { and } g(x)=\frac{4}{1+e^{-2 x}} $
Define the composite function $f^{\circ} g^{-1}$ by $\left(f^{\circ} g^{-1}\right)(x)=f\left(g^{-1}(x)\right)$, where $g^{-1}$ is the inverse of the function $g$.
Then the value of the derivative of the composite function $f^{\circ} g^{-1}$ at $x=2$ is $\_\_\_\_$ i