Explain the minus sign in the following formula.
As $z=f(x,y)=c$ $$\bigtriangledown z=\frac{\partial z}{\partial x}dx+\frac {\partial z}{\partial y}dy=0$$ so$$ \frac {dy}{dx}=-\frac{\frac{\partial z}{\partial x}}{\frac {\partial z}{\partial y}} $$
As $z=f(x,y)=c$ $$\bigtriangledown z=\frac{\partial z}{\partial x}dx+\frac {\partial z}{\partial y}dy=0$$ so$$ \frac {dy}{dx}=-\frac{\frac{\partial z}{\partial x}}{\frac {\partial z}{\partial y}} $$