Summary: The equality \(\sup_p {{|p' (a)|} \over {|p |_{[a, b]}}}= {{2n^2} \over {b-a}}\) is shown, where the supremum is taken for all exponential sums \(p\) of the form \(p(t)= a_0+ \sum_{j=1}^n a_j e^{\lambda_j t}\), \(a_j\in \mathbb{R}\), with nonnegative exponents \(\lambda_j\). The inequalities \[ \begin{aligned} |p' |_{[a+ \delta, b- \delta]} &\leq 4(n+ 2)^3 \delta^{-1} |p|_{[a, b]}\\ \text{and} |p' |_{[a+ \delta, b- \delta]} &\leq 4\sqrt {2} (n+ 2)^3 \delta^{-3/2} |p|_{L_2 [a, b]} \end{aligned} \] are also proved for all exponential sums of the above form with arbitrary real exponents. These results improve inequalities of Lorentz and Schmidt and partially answer a question of Lorentz.