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(x)=\log _e (x^2+2 x+4), and g(x)=\frac{4}{1+e^{-2 x}}. Define the composite function f \circ g^{-1} by (f \circ g^{-1}) (x)=f (g^{-1}(x)), 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 Adv 2025 P2]

Functions

Section: Mathematics

Question : 2 of 30

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Let R denote the set of all real numbers.
Let f:R⟶R and g:R⟶(0,4) be functions defined by f(x)=loge(x2+2x+4), and g(x)=‌

1+e−2x

.
Define the composite function f∘g−1 by (f∘g−1)(x)=f(g−1(x)), where g−1 is the inverse of the function g.
Then the value of the derivative of the composite function f∘g−1 at x=2 is ____ .

[JEE Adv 2025 P2]

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