We present a model-independent study of \(c \rightarrow d \mu ^{+} \nu _{\mu }\) transitions to search for new physics in the presence of right-handed neutrinos. We have adopted the effective field theory approach for the low-energy effective Hamiltonian comprising the dimension-six operators. The New Physics Wilson coefficients are determined through a \(\chi ^2\) fit by using the Miniut package to available experimental data of leptonic \(D^{+} \rightarrow \mu ^+ \nu \) and semileptonic decays \(D^{0} \rightarrow \pi ^{-} \mu ^+ \nu \) , \(D^{0} \rightarrow \rho ^{-} \mu ^+ \nu \) , \(D^{+} \rightarrow \rho ^0 \mu ^+ \nu \) . The differential decay width is derived to study the \(B_{c}^{+} \rightarrow B_{d} \mu ^+ \nu _{\mu }\) decay for the effect of right-handed neutrinos. We also make the predictions of \(q^2\) spectra and forward-backward asymmetry ( \(A_{FB}\) ) for the mode \(B_{c}^{+} \rightarrow B_{d} \mu ^+ \nu _{\mu }\) to inspect the effect of the allowed new physics in \( c \rightarrow d\) sector through right-handed neutrinos to motivate the future measurements.