In this work we analyze the main properties of the Zariski and maximal spectra of the ring S r (M ) of differentiable semialgebraic functions of class C r on a semialgebraic set M ⊂ R m . Denote S 0 (M ) the ring of semialgebraic functions on M that admit a continuous extension to an open semialgebraic neighborhood of M in Cl(M ). This ring is the real closure of S r (M ). If M is locally compact, the ring S r (M ) enjoys a Lojasiewicz's Nullstellensatz, which becomes a crucial tool. Despite S r (M ) is not real closed for r ≥ 1, the Zariski and maximal spectra of this ring are homeomorphic to the corresponding ones of the real closed ring S 0 (M ). In addition, the quotients of S r (M ) by its prime ideals have real closed fields of fractions, so the ring S r (M ) is close to be real closed. The missing property is that the sum of two radical ideals needs not to be a radical ideal. The homeomorphism between the spectra of S r (M ) and S 0 (M ) guarantee that all the properties of these rings that arise from spectra are the same for both rings. For instance, the ring S r (M ) is a Gelfand ring and its Krull dimension is equal to dim(M ). We also show similar properties for the ring S r * (M ) of differentiable bounded semialgebraic functions. In addition, we confront the ring S ∞ (M ) of differentiable semialgebraic functions of class C ∞ with the ring N (M ) of Nash functions on M .