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)`.

Explanation

We have f(x) g(x)=1. Differentiating with respect to x,
We get
`f'g)+fg'=0" (1)"`
Differentiating (1) w.r.t. x, we get
`f''g+2f'g'+fg''=0" (2)"`
Differentaintg (2) w.r.t x, we get
`f''g+g'''f+3f''g+3g''f'=0`
`"or (f''')/(f')(fg')+(fg')(g''')/(g')(f'g)+(3f'')/(f)(f'g)+(3g'')/(g)(fg')=0" [Using (1)]"`
`"or "((f''')/(f')+(3g'')/(g))(f'g)=-((g''')/(g')+(3f'')/(f))(f'g)`
`"or "-((f''')/(f')+(3g'')/(g))(fg')=((g''')/(g')+(3f'')/(g))fg'" [Using (1)]"`
`"or "(f''')/(f')+(3g'')/(g)=(g''')/(g')+(3f'')/(f)`
`"or "(f''')/(f')-(g''')/(g')=3((f'')/(f)-(g'')/(g))`