Let f : R → R be given by $f(x+y)=f(x)-f(y)+2 x y+1$ for all $x, y \in R$. If f(x) is everywhere differentiable and f'(0) = 1, then f'(x) =

$2 x-1$

We have,

$f(x+y)=f(x)-f(y)+2 x y+1$  or all  $x, y \in R$          ......(i)

Putting x = y = 0, we get

$f(0)=f(0)-f(0)+0+1 \Rightarrow f(0)=1$

Now,

$f'(x)=\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\Rightarrow f'(x)=\lim\limits_{h \rightarrow 0} \frac{f(x)-f(h)+2 x h+1-f(x)}{h}$

$\Rightarrow f'(x)=\lim\limits_{h \rightarrow 0}\left\{2 x-\frac{f(h)-1}{h}\right\}$

$\Rightarrow f'(x)=2 x-\lim\limits_{h \rightarrow 0} \frac{f(h)-f(0)}{h}=2 x-f'(0)=2 x-1$