If `f(x)=x^(3)+3x+4` and g is the inverse function of f(x), then the value of `(d)/(dx)((g(x))/(g(g(x))))` at x = 4 equals

Options

A. `(-1)/(6)`
B. 6
C. `(-1)/(3)`
D. non-existent

Explanation

`(d)/(dx)((g(x))/(g(g(x))))=(g(g(x)).g'(x) -g(x).g'(g(x)).g'(x))/(g(g(x))^(2))`
Now, f(0) = 4
`rArr" "f^(-1)(4)=g(4)=0`
`"and "g'(4)=(1)/(f'(0))=(1)/(3).`
Also, f(-1)=0
`rArr" "f^(-1)(0)=g(0)=-1`
`therefore" "(d)/(dx)((g(x))/(g(g(x))))_(x=4)`
`=(g(g(4))g'(4)-g(4)g'(g(4))g'(4))/((g(4))^(2))`
`=(g(0)cdot(1)/(f'(0))-0)/((g(0))^(2))`
`=(-1xx(1)/(3))/(1)`
`=(-1)/(3)`