The production of single top quarks in high energy collision events is a unique process to investigate the strength and structure of electroweak interactions. With the start of the physics program at the LHC, the production of single top quarks can be studied for the first time in great detail using an unprecedented quantity of proton-proton collision events. The presented work encompasses first measurements of differential single top quark cross sections in t-channel which have been performed using data collected at center-of-mass energies of 8 and 13 TeV with the CMS detector at CERN, Switzerland. The electroweak coupling structure of the standard model of particle physics predicts only left-handed top quarks or right-handed top antiquarks to participate in such interactions which results in the production of highly polarized top quarks. The degree of polarization is studied by measuring the cross section as a function of the polarization angle defined between the charged lepton from the top quark decay and a spectator quark which is produced in association with the single top quark. For the measurement events containing a muon, two or three jets and significant missing transverse energy, recorded at a center-of-mass energy of 8 TeV, are analyzed. The selected events are classified further with the help of two boosted decision tree discriminants that allow to suppress contributions from various background processes mimicking the single top quark signature. The differential cross section at parton-level as a function of the polarization angle is inferred from data events using an unfolding technique. The top quark spin asymmetry, a quantity related to the polarization, is extracted from the differential cross section. In a further step, the derivation of limits on potential anomalous top quark couplings occurring in new physics models is illustrated using the result of the measurement. In the second part, an early measurement of differential cross sections as a function of the top quark transverse momentum and rapidity is presented using the first data recorded at a center-of-mass energy of 13 TeV. For this, the analysis strategy of the top quark polarization measurement has been significantly extended which results in an improved estimation of the amount of single top quark events in data while contributions from background processes are profiled in-situ. The measured differential cross sections are compared to the predictions by various event generator programs.