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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.

A linear spline is a continuous function whose restriction to every interval $[x_{i-1},x_i]$ is a polynomial of polynomial degree at most one. With $h_i=x_i-x_{i-1}$ and ${\ell }_i=\ell (x_i)$, linear interpolation givesIntegrating the two linear polynomials and adding the contributions givesThis is the composite trapezoidal rule applied to the nodal values, and it is exact for the linear spline itself, including nonuniform spline knots.

A cubic spline on this partition of an interval is a function $S\in C^2[x_0,x_n]$ that restricts to a polynomial of polynomial degree at most three on every interval $[x_{i-1},x_i]$. Thus its second derivative is continuous and piecewise of polynomial degree at most one: $S^{\prime \prime }$ is a linear spline on the same spline knots. Its third derivative is constant on each open interval and can jump across the spline knots; its fourth derivative vanishes inside each open interval.

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.