Let `g(x)` be the inverse of an invertible function `f(x),` which is differentiable for all real `xdot` Then `g^(f(x))` equals. `-(f^(x))/((f^'(x))^3)` (b) `(f^(prime)(x)f^(x)-(f^(prime)(x))^3)/(f^(prime)(x))` `(f^(prime)(x)f^(x)-(f^(prime)(x))^2)/((f^(prime)(x))^2)` (d) none of these

Options

A. `-(f''(x))/((f'(x))^(3))`
B. `(f'(x)f''(x)-(f(x))^(3))/(f'(x))`
C. `(f'(x)f''(x)-(f'(x))^(2))/((f'(x))^(2))`
D. none of these

Explanation

`"Given that "g^(-1)(x)=f(x)`
`"or "x=g(f(x))`
`"or "g'(f(x))f'(x)=1`
`"or "g'(f(x))=(1)/(f'(x))`
`"or "g''(f(x))f'(x)=(-f''(x))/([f'(x)]^(2))`
`"or "g''(f(x))=(-f''(x))/([f'(x)]^(3))`