Let `f: R-{0}rarrR` be a function which is differentiable in its domain and satisfying the equation `f(x+y)=f(x)+f(y)+(x+y)/(xy)-(1)/(x+y),` also f'(1)=2. Then find the function.

Explanation

`f(x+y)=f(x)+f(y)+(1)/(x)+(1)/(y)-(1)/(x+y)`
Differentiating w.r.t. to x, keeping y as constant.
`f'(x+y)=f'(x)-(1)/(x^(2))+(1)/((x+y)^(2))` Putting x= 1
`f'(x+y)=f'(x)-(1)/(x^(2))+(1)/((x+y)^(2))`
`rArr" "f'(x)=1+(1)/(x^(2))`
`rArr" "f(x)=x-(1)/(x)+c`
Put x = y=1 in (1), we get
`f(2)-2f(1)=2-(1)/(2)=(3)/(2)`
`"Also from "(2),f(2)=2-(1)/(2)-C`
`rArr" "C=0`
`rArr" "f(x)=x-(1)/(x)`