Question

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

• A. g(x)
• B. l +g(x)
• C. $1+(g-(x)^3$
• D. $\frac{1}{1+\left(g\left(x\right)\right)^{3}}$
• E. 0

Answer 1

Answer: (C)

Given, $g(x)=f^{-1}(x)$
$\Rightarrow \, f\{g(x)\}=x$
On differentiating, w.r.t. $x$, we get
$ f'\{g(x)\} \cdot g'(x)=1 $
$\Rightarrow \, g'(x)=\frac{1}{f'\{g(x)\}}$
$ \because \, f'(x)=\frac{1}{1+x^{3}} $ (given)
$\therefore \, f'\{g(x)\}=\frac{1}{1+\{g(x)\}^{3}}$
Now, from Eq. (i), we get
$g'(x)=1+\{g(x)\}^{3}$