Let `f(x)a n dg(x)` be two function having finite nonzero third-order derivatives `f'''(x)` and `g'''(x)` for all `x in R`. If `f(x).g(x)=1` for all `x in R ,` then prove that `(f''')/(f') - (g''')/(g') = 3((f'')/f - (g'')/g)`.

Options

A. `(-2f'(x))/(f(x))`
B. `(-2g'(x))/(g(x))`
C. `(-f'(x))/(f(x))`
D. `(2f'(x))/f(x)`

Explanation

`"We have "=(1)/(f)`
`therefore" "g'=(-1)/(f^(2))f'`
`"or "g''=-[-(2)/(f^(3))f'^(2)+(1)/(f^(2))f'']`
`=(2)/(f^(3))f'^(2)-(f'')/(f^(2))`
`"or "(f'')/(f')-(g'')/(g')=(f'')/(f')-((2)/(f^(3))f'^(2)-(f'')/(f^(2)))/(-(1)/(f^(2))f')`
`=(f'')/(f')-((-2f')/(f)+(f'')/(f'))=(2f')/(f)`
`"Alos, "g cdotf=1`
`"or "g'f+gf'=0`
`therefore" "(f'')/(f')-(g'')/(g')=-(2g')/(g)`