Third derivative ($f^{\prime \prime \prime }=\frac{d}{dx}{f}^{\prime \prime }$)

The third derivative is the derivative of the second derivative. On the interior of a cubic polynomial piece it is constant. A cubic spline can have different constants on adjacent pieces, so its third derivative need not be continuous across a spline knot.

The fourth derivative is the derivative of the third derivative. It vanishes for a polynomial of polynomial degree at most three. For a cubic spline, this statement applies inside each knot interval; a jump of the third derivative can prevent a classical fourth derivative from existing at a spline knot.