In the dynamic minimum set cover problem, the challenge is to minimize the update time while guaranteeing a close-to-optimal min\{O(log n), f\} approximation factor. (Throughout, n, m, f, and C are parameters denoting the maximum number of elements, the number of sets, the frequency, and the cost range.) In the high-frequency range, when f = \Omega(log n), this was achieved by a deterministic O(log n)-approximation algorithm with O(f log n) amortized update time by Gupta et al. [Online and dynamic algorithms for set cover, in Proceedings STOC 2017, ACM, pp. 537-550]. In this paper we consider the low-frequency range, when f = O(log n), and obtain deterministic algorithms with a (1 + \epsilon)f-approximation ratio and the following guarantees on the update time. (1) O ((f/\epsilon) \cdot log(Cn)) amortized update time: Prior to our work, the best approximation ratio guaranteed by deterministic algorithms was O(f2) of Bhattacharya, Henzinger, and Italiano [Design of dynamic algorithms via primal-dual method, in Proceedings ICALP 2015, Springer, pp. 206-218]. In contrast, the only result with O(f)-approximation was that of Abboud et al. [Dynamic set cover: Improved algorithms and lower bounds, in Proceedings STOC 2019, ACM, pp. 114-125], who designed a randomized (1 + \epsilon)f-approximation algorithm with O((f2 /\epsilon) \cdot log n) amortized update time. (2) O \bigl(f2 /\epsilon3 + (f/\epsilon2) \cdot log C\bigr) amortized update time: This result improves the above update time bound for most values of f in the low-frequency range, i.e., f = o(log n). It is also the first result that is independent of m and n. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [Deterministically maintaining a (2 + \epsilon)-approximate minimum vertex cover in O(1/\epsilon2) amortized update time, in Proceedings SODA 2019, SIAM, pp. 1872-1885] for unweighted dynamic vertex cover (i.e., when f = 2 and C = 1). (3) O((f/\epsilon3) \cdot log2(Cn)) worst-case update time: No nontrivial worst-case update time was previously known for the dynamic set cover problem. Our bound subsumes and improves by a logarithmic factor the O(log3 n/\sansp\sanso\sansl\sansy(\epsilon)) worst-case update time for the unweighted dynamic vertex cover problem (i.e., when f = 2 and C = 1) of Bhattacharya, Henzinger, and Nanongkai [Fully dynamic approximate maximum matching and minimum vertex cover in O(log3)n worst case update time, in Proceedings SODA 2017, SIAM, pp. 470-489]. We achieve our results via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. Prior work in dynamic algorithms that employs the primal-dual approach uses a local update scheme that maintains relaxed complementary slackness conditions for every set. For our first result we use instead a global update scheme that does not always maintain complementary slackness conditions. For our second result we combine the global and the local update schema.