$\text{ If }f:R\to R\text{ be a differentiable function, such that }$$f(x+2y)=f\left(x\right)+f\left(2y\right)+4xy\text{ for all }x,y\in R\text{ then }$

$f(x+2y)=f\left(x\right)+f\left(2y\right)+4xy\text{ for }x,y\in R\text{ putting }x=y=0\text{, we }$

$\begin{array}{l}\text{ get }f\left(0\right)=0\text{ now, }f(x+2y)=f\left(x\right)+f\left(2y\right)+4xy\\ \Rightarrow \frac{f(x+2y)-f\left(x\right)}{2y}=2x+\frac{f\left(2y\right)}{2y}\\ \Rightarrow \underset{y\to 0}{lim} \frac{f(x+2y)-f\left(x\right)}{2y}=\underset{y\to 0}{lim} \left\{2x+\frac{f\left(2y\right)-f\left(0\right)}{2y}\right\}\\ \\ =\underset{y\to 0}{lim} \left(2x+\frac{f\left(2y\right)-0}{2y}\right)\\ \Rightarrow f^{\mathrm{\prime }}\left(x\right)=2x+f^{\mathrm{\prime }}\left(0\right)\text{ for all }X\\ \Rightarrow f^{\mathrm{\prime }}\left(1\right)=2+f^{\mathrm{\prime }}\left(0\right)\end{array}$