Sometimes, it is impossible to give the equation of a curve in the form

for example, the set of $(x,y)$ satisfying

gives us the curve

but we cannot give $y$ as an explicit function of $x.$ Rather, the points $y$ are defined implicitly.

In this case, we still imagine $y$ is a function of $x,$ and to differentiate

we use the chain rule:

1. The curve $C$ is defined by

   $$x{y}^2-2x+y^2+y-3=-4$$

   Verify that the point $P(-1,-3)$ lies on the curve, and find the rate of change of $y$ with respect to $x$ at this point.

2. Find the equation of the tangent to the curve

   $$xy-3x-2y+5=-2$$

   at the point where $x=3$.