When is the derivative of a quadratic spline a spline itself?

Question

Let the quadratic spline $s_2$ interpolate $f\in\mathcal{C}^2([a,b])$ in the points $a=x_0<x_1<\cdots<x_n=b$. Is $s_2'(x)$ a linear spline?

$s_2'$ is certainly a linear function and continuous, but without assuming it, it might not interpolate $f'$ in any points, right?

Answer 1

Yes, my supposition is correct.