The Chain Rule & Implicit Differentiation · Challenge problem

A differentiable function $f(x,y)$, defined on the half-plane $y > 0$, satisfies $f(t, t^2) = t^4 + 1$ and $f(t^2, t) = 2t^3$ for every $t > 0$. Let $g(t) = f(t, t)$. Find $g'(1)$.