In this applet you can define an implicit curve (limited to polynomial equation in variables x and y). The red point C can be dragged along the curve; when C is at coordinates (a,b) the green point P has coordinates (a,m), where m is the slope of the curve at the point C=(a,b).

Drag C along the curve--you can even drag it to another piece of the curve--and watch as P leaves a green trace recording the slope at each point you've visited. In particular note that the slope depends on both x and y, so that two points with the same x coordinate can have very different slopes.
For the equation defining your curve use implicit differentiation to find dy/dx. Place C and compute dy/dx at this point. Does your result agree with the second coordinate of P?