If $p(x)$ is polynomial such that $$p(x^2+1)=(p(x))^2 +1$$ and $p(0)=0$ then $p'(0)$ is equal to _____.
While looking on it and putting $x=0$ I got $p(1)=1$ and also the equation given had also the similar form (in comparison) of $$y^2+1=(y^2)+1$$. So, I put this polynomial in test and found that it satisfies this as $p(x^2+1)=x^2+1$ and and $p(x)=x$ and hence it satisfied for every real x and so $p'(0)=1$. But I was not able to progress in the direction of correct solving.
Also, the given related problem is there on this site but, I have not been taught the coefficient determination and hence that answer there is out of my scope for now. But,I think there is this derivative method which I am currently learning and can understand about it.So, I would like to know how to approach this in derivative method and also if there is any other easier way without proving $p(x)$ to be the unique solution.
Also, it may happen that if any other exists,then many persons will get to know about the new method. It will be more helpful to them and me inlcuded.
So, is there any hints or suggestions on this problem?
Thanks for the help!