The increasingly large number of complex organic molecules detected in the interstellar medium necessitates robust kinetic models that can be relied upon for investigating the involved chemical processes. Such models require rate coefficients for each of the thousands of reactions; the values of these are often estimated or extrapolated, leading to large uncertainties that are rarely quantified. We have performed a global Monte Carlo and a more local one-at-a-time sensitivity analysis on the gas-phase rate coefficients in a 3-phase dark cloud model. Time-dependent sensitivities have been calculated using four metrics to determine key reactions for the overall network as well as for the cyanonaphthalene molecule in particular, an important interstellar species that is severely under-produced by current models. All four metrics find that reactions involving small, reactive species that initiate hydrocarbon growth have large effects on the overall network. Cyanonaphthalene is most sensitive to a number of these reactions as well as ring-formation of the phenyl cation (C6H5+) and aromatic growth from benzene to naphthalene. Future efforts should prioritize constraining rate coefficients of key reactions and expanding the network surrounding these processes. These results highlight the strength of sensitivity analysis techniques to identify critical processes in complex chemical networks, such as those often used in astrochemical modeling.