Let \(f(x)\) be a differentiable function defined on \([-1,1]\). One of the most powerful procedure for the numerical calculation of the derivative \(f^\prime (x)\) is given by the Chebyshev spectral collocation process, that is by taking the derivative of an interpolating polynomial. As the authors note, the Chebyshev differencing method works best when the function \(f(x)\) can be continued into the complex plane as a function \(f(z)\) which is analytic in an open neighborhood of \([-1,1]\). In this case, an upper bound of the error, in the discrete maximum norm, can be derived. The main result of the paper states that, if \(p_N(x)\) is the polynomial interpolant of \(f(x)\) at the set of zeros \(\{ s_j \} \) or extrema \(\{ t_j \} \) of the Chebyshev polynomial \(T_N (x)\), and \(f(z)\) is analytic in an ellipse containing the interval \([-1,1]\), then \(\max _{0 \leq j \leq N}| f^\prime (s_j)- p^\prime (s_j)| \) and \(\max _{0 \leq j \leq N}| f^\prime (t_j)- p^\prime (t_j)| \) are bounded and their bounds are given explicitly. Two model functions are analyzed in detail.