Tangents of the lower half of a circle

The bottom half of a circle has tangents that don't look anything like the tangents of a parabola or the trough of a sine wave. Most notably, the last two tangent lines, the one on the left and the one on the right, are both vertical.

Both the parabola and the circle have tangents that are neatly spaced out, but the parabola seen here only has slopes that range from -2 to 2, while the circle has slopes from negative infinity to infinity.

Tomorrow, a somewhat lesser known curve sometimes mistaken for a parabola, the catenary.