Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.