We consider derivatives of the eigenfunctions and associated functions of the formal Sturm--Liouville operator $$ \Cal L(u)(x)=-\bigl(p(x)u'(x)\bigr)'+q(x)u(x) $$ defined on a finite or infinite interval $G\subseteq\Bbb R$. We suppose that the complex-valued potential $q=q(x)$ belongs to the class $L_1^{\text{\rm loc}}(G)$ and that piecewise continuously differentiable coefficient $p=p(x)$ has a finite number of the discontinuity points in $G$. Order-sharp upper estimates are obtained for the suprema of the moduli of the first derivative of the eigenfunctions and associated functions of the operator $\Cal L$ in terms of their norms in metric $L_2$ on compact subsetes of $G$ (on the entire interval $G$).