Implicit function derivation

Question

I have function $h(x,y)=e^{xy^2-1}+\log{\frac{x}{y}}-1$ and I have to find if a function $y=f(x)$ around $[1,1]$ exists. I have to check some conditions in order to find out if $y=f(x)$, $h(1,1)=0,\frac{\partial h}{\partial y}(1,1)\not =0$ and finally wheter $h(x,y)\in C^1(U)$,where $U$ is neighbourhood(not sure if it is correct term in English) $U=B([1,1],\epsilon )$. Since I want to be safe and I don't want to think too much during test, can I choose really small $\epsilon$ so the $\frac{x}{y}$ in logarithm is not negative or zero?

Answer 1

Set $x=1+\epsilon u$ and $y=1+\epsilon v$ and expand $h(1+\epsilon u,1+\epsilon u)$ in $\epsilon$:

$$h(1+\epsilon u,1+\epsilon u)=(2u+v)\epsilon+O(\epsilon^2)$$

Thus we obtain:

$$v=-2u \implies (y-1)=-2(x-1) \implies y(x)=-2x+3$$.