The naive summation of a Chebyshev series \begin{align*} f(x) = \frac{c_0}{2} + \sum_{k=1}^{n-1} c_{k}T_{k}(x) \end{align*} which employs the three-term recurrence for evaluation of the Chebyshev polynomials is known to be numerically unstable near $x=\pm 1$. For this reason, Clenshaw presented an algorithm which avoids this instability; see Clenshaw's original work and an exposition in Algorithm 3.1 of Numerical Methods for Special Functions.
Is there literature which presents a numerically stable method for evaluating the derivative of a Chebyshev series, perhaps using an analogy to the Clenshaw recurrence presented above? Or is the naive method stable?