What conic section is a hyperbola derived from?
It is not derived from a conic section - it is a conic section.
Given a cone in three dimensional space, if we take a plane slice of it, then we can get any of the following curves where the plane and the cone intersect:
Circle. This happens when the plane is perpendicular to the axis of the cone.
Ellipse. This happens when the plane is tilted from the perpendicular but less than the angle of the cone.
Parabola. This happens when the plane is at the same angle to the axis as the cone itself.
Hyperbola. This happens when the plane is at a smaller angle to the axis than the cone itself.
There are various exceptions resulting in a single point, line or pair of lines, but otherwise, the above are the possible conic sections.