Using the dynamic mean-field approximation of the totally asymmetric simple exclusion process (TASEP), we investigate the effect of small changes in the initiation, elongation, and termination rates along the mRNA strand on the steady-state protein translation rate. We show that the sensitivity of mRNA translation is equal to the sensitivity of the maximal eigenvalue of a symmetric, nonnegative, tridiagonal, and irreducible matrix. This leads to new analytical results as well as efficient numerical schemes that are applicable for large-scale models. Our results show that in the usual endogenous case, when initiation is more rate-limiting than elongation, the sensitivity of the translation rate to small mutations rapidly increases towards the 5' end of the ORF. When the initiation rate is high, as may be the case for highly expressed and/or heterologous optimized genes, the maximal sensitivity is with respect to the elongation rates at the middle of the mRNA strand. We also show that the maximal possible effect of a small increase/decrease in any of the rates along the mRNA is an increase/decrease of the same magnitude in the translation rate. These results are in agreement with previous molecular evolutionary and synthetic biology experimental studies.