Question

If $g$ is the inverse of $f$ and $f^{\prime }\left(x\right)=\frac{1}{1+x^3}$, then $g^{\prime }\left(x\right)$ is equal to.

• A. $1+\left[g\right(x){]}^3$
• B. $\frac{1}{1+\left[g\right(x){]}^3}$
• C. $\left[g\right(x){]}^3$
• D. $Noneofthese$

Answer 1

Answer: (A)

$1+\left[g\right(x){]}^3$

Here, $f^{\prime }\left(x\right)=\frac{1}{1+x^3}$

$\Rightarrow$ $g$ is inverse of function $f$, then

$f\left[g\right(x\left)\right]=x$

Differentiating w.r.t $x,$

$f^{\prime }\left[g\right(x\left)\right]\times g^{\prime }\left(x\right)=1$

So $g^{\prime }\left(x\right)=\frac{1}{f^{\prime }\left[g\right(x\left)\right]}$

$\therefore$ $g^{\prime }\left(x\right)=1+\left[g\right(x){]}^3$