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Q.
If $f(x)=x^3+3 x+4$ and $g$ is the inverse function of $f$ then the value of $\frac{d}{d x}\left(\frac{g(x)}{g(g(x))}\right)$ at $x=4$ equals
Continuity and Differentiability
Solution:
$\frac{ d }{ dx }\left(\frac{ g ( x )}{ g ( g ( x )))}\right)=\frac{ g ( g ( x )) g ^{\prime}( x )- g ( x ) \cdot g ^{\prime}( g ( x )) g ^{\prime}( x )}{( g ( g ( x )))^2}$
Now, $f ( x )= x ^3+3 x +4$
$f(0)=4$ OR $g(4)=0$
$g^{\prime}(4)=\frac{1}{f^{\prime}(0)}=\frac{1}{3} $
$f(-1)=0 \Rightarrow g(0)=-1,$
Put at $x =4$
$\frac{ d }{ dx }\left(\frac{ g ( x )}{ g ( g ( x ))}\right)=\frac{ g ( g (4)) \cdot g ^{\prime}(4)- g (4) \cdot g ^{\prime}( g (4)) g ^{\prime}(4)}{( g ( g (4)))^2}=\frac{(-1)\left(\frac{1}{3}\right)-0}{1}=-\frac{1}{3}$