along the line y = x we have to have dy/dx = -1 if the curve is symmetric, do we not?

think of it this way: suppose our curve WERE symmetric and is a point of the curve below y = x, and is the corresponding point of the curve above y = x.

if the curve is symmetric, we have .

now, what is the slope between those two points? it's:

now suppose a point where the curve crosses y = x is (a,a). then as , the slope between the two points approaches dy/dx, but this slope is constantly -1.