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 $\_\_\_\_$.
[0.2 to 0.3]
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 $\_\_\_\_$.