Problem

(3) (10 pts) Let $f:(-\delta,\delta)\to\mathbb{R}$ be a function so that $f(0)=\dfrac{1}{2}$ for $\delta>0$. Assume that $f(x)$ satisfies the equation

$$x^2+\bigl(f(x)\bigr)^2=\bigl(2x^2+2\bigl(f(x)\bigr)^2-x\bigr)^2$$

for all $x\in(-\delta,\delta)$. Suppose we know that $f$ is differentiable at $0$. Compute the tangent line to the curve $y=f(x)$ at the point $\left(0,\dfrac{1}{2}\right)$.