$(ln(x))(ln(y)) = ln(xy)$ in point $ (1,1)$
I want to know if my answer is correct or if my thinking is wrong and the answer is finding using another way. For the equation of the line, we have the formula: $y- y_{0} = m(x - x_{0})$. Now We need to find the inclination of the curve: \begin{align} \frac{d}{dx}[ln(x)*ln(y)] = \frac{d}{dx}[ln(xy)]\\ \end{align} \begin{align} ln(y) \frac{d}{dx}[ln(x)]+ln(x)\frac{d}{dx}lny = \frac{1}{xy}\frac{d}{dx}\end{align} \begin{align} \frac{d}{dx}[ln(x)-1/xy]= -\frac{lny}{x} \end{align} \begin{align} \frac{dy}{dx}= \frac{-ln(y)xy^2}{x^2yln(x)-y} \end{align}
But ln(1) = 0, then $\frac{dy}{dx}=0$. In conclusion, my equation of the line is: $y=y_{0}$. So, $y = 1$?