functional analysis - $f(x)$ if $f(xy)=f(x) +f(y) +frac{x+y-1}{xy}$

Let $f$ be a differentiable function satisfying the functional time $f(xy)=f(x) +f(y) +\frac{x+y-1}{xy} \forall x,y \gt 0 $ and $f'(1)=2$

My work

Putting $y=1$

$$f(1)=-1$$
$$f'(x)=\lim_{h\to 0}\frac{f(x+h) - f(x)}{h}$$

But I don't know anything about $f(x+h)$ so what to do in this problem ?

Answer

Differentiate both sides with respect to $x$:
$$
yf'(xy)=f'(x)-\frac{1}{x^2}+\frac{1}{x^2y}
$$
For $y=1/x$, we get
$$
\frac{f'(1)}{x}=f'(x)-\frac{1}{x^2}+\frac{1}{x}
$$

so
$$
f'(x)=\frac{1}{x^2}+\frac{f'(1)-1}{x}
$$