Question

If $g$ is the inverse of a function $f$ and $f^{\prime}(x)=\frac{1}{1+x^5}$, then $g^{\prime}(x)$ equal to:

(1) $\frac{1}{1+\{g(x)\}^5}$

(2) $1+\{g(x)\}^5$

(3) $1+x^5$

(4) $5 x^4$

Answer 1

SOLUTTION : If $f(x) \& g(x)$ are inverse of each other then,

$g^{\prime}(f(x))=\frac{1}{f^{\prime}(x)} ; \quad g^{\prime}(f(x))=1+x^5 \\$

$\text { Here } x=g(y) \Rightarrow \quad g^{\prime}(y)=1+[g(y)]^5 \quad \Rightarrow \quad g^{\prime}(x)=1+\{g(x)\}^5$

Above Correct Answer is Option (2)