Suppose a differentiable function

$f(x+y)=f(x)+f(y)+x y^{2}+x^{2} y$

Differentiate w.r.t. $x$ :

$f^{\prime}(x+y)=f^{\prime}(x)+0+y^{2}+2 x y$

Put $y=-x$

$f^{\prime}(0)=f^{\prime}(x)+x^{2}-2 x^{2}$....(i)

$\because \lim _{x \rightarrow 0} \frac{f(x)}{x}=1 \Rightarrow f(0)=0$

$\therefore f^{\prime}(0)=1$.....(ii),

From equations (i) and (ii),

$f^{\prime}(x)=\left(x^{2}+1\right) \Rightarrow f^{\prime}(3)=10$