Let $g: \mathbb R^{+} \times \mathbb R^{+}\to \mathbb R^{2}\setminus\{0\}$, $g(x,y):=(xy,\sqrt{x}/y)$
and
$f: \mathbb R^{2}\setminus\{0\}\to \mathbb R$, $f(x,y):=\ln(x^{2}y^{2})$
Find the Taylor Series of $h: f\circ g$ about $(x_{0},y_{0})$ to the second order.
My idea: After just having completed one-dimensional Taylor Series in our lecture, I would think that I could find the Taylor Polynomial of $f$ and then plug in my values $(g(x_{0},y_{0}))$ into the existing Taylor Polynomial of $f$ in order to get the $f \circ g$. This would be analogous to the one-dimensional Taylor Polynomial of a composite function (e.g. If I wanted to find the Taylor Polynomial of $e^{x^{3}-5x+1}$ about $0$, I would use the Taylor polynomial of $e^{x}$ about $0$ and then simply plug in $x^{3}-5x+1$( Correct me if I am wrong).
Why does this principle not work in multiple dimensions?