Q67 / 155 · WBJEE 2025 · Mathematics

If ' $f$ ' is the inverse function of ' $g$ ' and $g^{\prime}(x)=\frac{1}{1+x^n}$, then the value of $f^{\prime}(x)$ is

  • Correct answer
    A
    $1+\{f(x)\}^n$
  • B
    $1-\{f(x)\}^n$
  • C
    $\{1+f(x)\}^n$
  • D
    $\{f(x)\}^n$

Solution

$\begin{aligned} & g^{-1}(x)=f(x) \\ & g^{\prime}(x)=\frac{1}{1+x^n} \quad \Rightarrow g^{\prime}\left(g^{-1}(x)\right)=\frac{1}{1+\left(g^{-1}(x)\right)^n}=\frac{1}{1+(f(x))^n} \\ & g\left(g^{-1}(x)\right)=x \\ & g^{\prime}\left(g^{-1}(x) \cdot\left(g^{-1}(x)\right)^{\prime}=1\right. \\ & \left(g^{-1}(x)\right)^{\prime}=\frac{1}{g^{\prime}\left(g^{-1}(x)\right)}=1+(f(x))^n\end{aligned}$

Hence, the correct answer is option: 'A'.

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