Let $R$ denote the set of all real numbers. Let $f: R \rightarrow R$ and $g: R \rightarrow(0,4)$ be functions defined by $f\left(x\right)={\log }_{e}\left({x}^{2}+2x+4\right),\text{ and }g\left(x\right)=\frac{4}{1+{e}^{-2x}}$ 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________. [JEE Advanced 2025]

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YESPYQ · Accepted Answer

Correct answer: c) 0.25. $(f\circ g^{-1})'(2)=f'(g^{-1}(2))(g^{-1})'(2).$ Since $g(0)=2$ $g^{-1}(2)=0$ $f'(0)=\frac{1}{2}$ and $g'(0)=2$ $(g^{-1})'(2)=\frac{1}{g'(0)}=\frac{1}{2}$ $(f\circ g^{-1})'(2)=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}.$

Let \(R\) denote the set of all real numbers. Let \(f: R \rightarrow R\) and \(g: R \rightarrow(0,4)\) be functions defi…

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