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Q.
Let $f( x )=\int\limits_3^{ x } \frac{ dt }{\sqrt{ t ^4+3 t ^2+13}}$. If $g ( x )$ is the inverse of $f( x )$ then $g ^{\prime}(0)$ has the value equal to
Integrals
Solution:
$ f ^{\prime}( x ) \frac{ dy }{ dx }=\frac{1}{\sqrt{ x ^4+3 x ^2+13}}$
$\therefore g ^{\prime}( y )=\frac{1}{ dy / dx }=\sqrt{ x ^4+3 x ^2+13}$
when $y=f(x)$
when $y=0$ then $x=3$
hence $g^{\prime}(0)=\sqrt{3^4+27+13}=\sqrt{121}=11$.