Differentiation question bank (Maths)

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 ________ .

Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x} f(x)=2+\int_0^x \sqrt{t^4+1} d t, \quad$ for all $x \in(-1,1)$ and let $f^{-1}$ be the inverse function of $f$. Then $\left(f^{-1}\right)^{\prime}(2)$ is equal to