Let ℝ denote the set of all real numbers. Let f: ℝ arrow ℝ and g: ℝ...

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 $f(x)=\log _e\left(x^2+2 x+4\right)$, 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)=\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 ________ .