If $ f(x.y) = f(x) + f(y) + (x+y-1)/xy $ then what is f(x)?

Question

$$ f(x.y) = f(x) + f(y) + (x+y-1)/xy $$ And it is given that $$ f'(1) = 2 $$ And we have to find $$ \lfloor f(e^{100})\rfloor $$ I tried to guess the function by some observation and deduces it to be as $$ f(x) = -1/x$$ but the thing is that it doesnt satisfy the second condition of the question .

Answer 1

Because of the product to sum property, you'll need a $\ln x$ in there. And since $$\frac{x+y-1}{xy} = \frac{1}{x}+\frac{1}{y} -\frac{1}{xy}$$ you'll probably want a $1/x$. And that seems to be enough. Let $f(x) = \ln x -1/x$. In fact, for any constant $a$, $f(x) = a\ln x - 1/x$ would solve the equation, and then you can adjust the $a$ to satisfy the derivative requirement. (Your solution uses $a=0$.)