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  4. If f ( x )= x 3+ ln x +1 and g ( x ) is the inverse function of f ( x ) then ( g prime(2))-1 is equal to

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Q. If $f ( x )= x ^3+\ln x +1$ and $g ( x )$ is the inverse function of $f ( x )$ then $\left( g ^{\prime}(2)\right)^{-1}$ is equal to

Continuity and Differentiability

Solution:

$f(x)=x^3+\ln x+1 \Rightarrow f^{\prime}(x)=3 x^2+\frac{1}{x} $
$f(1)=2$
$g^{\prime}(2)=\frac{1}{f^{\prime}(1)}=\frac{1}{4} \Rightarrow\left(g^{\prime}(2)\right)^{-1}=4$