JEE Advanced 2025 · Paper 2, Mathematics question 14

Let $\mathbb{R}$ denote the set of all real numbers. Let $f : \mathbb{R} \to \mathbb{R}$ and $g : \mathbb{R} \to (0, 4)$ be functions defined by

$$f(x) = \log_e\left(x^2 + 2x + 4\right),\quad\text{and}\quad g(x) = \frac{4}{1 + e^{-2x}}.$$

Define the composite function $f\circ g^{-1}$ by $(f\circ g^{-1})(x) = f\left(g^{-1}(x)\right)$, 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 _____.