Let `f(x)=int_(0)^(x)(dt)/(sqrt(1+t^(3)))` and `g(x)` be the inverse of `f(x)`. Then the value of `4 (g''(x))/(g(x)^(2))` is________.

Answer

6

Explanation

`y=f(x)impliesx=f^(-1)(y)impliesx=g(y)`
Given `y=f(x)=int_(0)^(x)(dt)/(sqrt(1+t^(3))`
`(dy)/(dx)=1/(sqrt(1+x^(3))` or `(dx)/(dy)=sqrt(1+x^(3))`
`g'(y)=sqrt(1+g^(3)(y))`
`g''(y)=(3g^(2)(y)g'(y))/(2sqrt(1+g^(3)(y)))`
`:. 2g''(y)=3g^(2)(y)(g'(y))/(sqrt(1+g^(3)(y)))=3g^(2)(y)(sqrt(1+g^(3)(y)))/(sqrt(1+g^(3)(y)))=3g^(2)(y)`
or `2g''(y)=3g^(2)(y)`