So suppose we have $G(z+h,w+t)$ where $G$ is an algebraic equation such that $G(z,w)=g_0(z)+g_1(z)w+\cdots+g_m(z)w^m$, each $g_i(z)$ is a polynomial of $z \in \mathbb{C}$, and $w$ is any function.
My textbook is telling me in a proof that $G(z+h,w+t)=G(z,w)+hG_z(z,w)+tG_w(z,w)+O(h^2+ht+t^2)$.
Can anyone elaborate why this is the case? Trying to expand does not seem fruitful to me.