In a quadratic

The discriminant is equal to the first derivative.

Does that have any special meaning (geometrically or otherwise), or is it just an algebraic coincidence?

the quadratic formula is -b/2a (+/-) sqrt(b^2 - 4ac)/ 2a = x

-b/2a is the vertex of the parabola.  The sqrt(b^2 - 4ac) is the distance on each side (Hence +/-) from the vertex to the zero's.

The vertex of a parabola exists when the first derivative is zero.   2ax + b = 0 when X = -b/2a.  Hence the relationship.