So I am taking mathematical economics and in the HW my professor asked to draw a couple of level curves for $f(x,y)=xy$.
Attempt: So I did it the following way. To find the slope I took the partial derivative with respect to $x$ and then with respect to $y$.:
$\frac{\partial f}{\partial x}=y\\\frac{\partial f}{\partial y}=x$
Now to find $\frac{\partial y}{\partial x}$ I just divide the upper equality $(\frac{\partial f}{\partial x})$ by the lower equality ($\frac{\partial f}{\partial y}$):
$\frac{\partial f}{\partial x} \frac{\partial y}{\partial f}=\frac{\partial y}{\partial x}=\frac{y}{x}$
So why is it wrong and why should I use the total differential to find the correct slope? Any hints please.