What does the derivative of a quadratic represent?

The vertex of a quadratic function lies at the zero of the derivative (which is linear)

If the slope of the derivative is negative, the function has a maximum. Otherwhise it has a minimum.

The zeros of the quadratic mean nothing in the derivative.

When the derivative of any function is positive, the function is increasing. When it's negative, it's decreasing. A linear function has both positive and negative values, negative in a range, positive in another range, so the quadratic has the corresponding decreasing and increasing parts, ranges.