We study the right-definite separated half-linear Sturm–Liouville eigenvalue problems. It is proved that the $n$th real eigenvalue of the problem depends smoothly on the equation, but may have jump discontinuities with respect to the boundary condition. Formulae are found for the derivatives of the $n$th real eigenvalue with respect to all parameters: the endpoints, the boundary condition and the coefficient functions, whenever they exist. Monotone properties and a comparison result for real eigenvalues are deduced as consequences. The generalized Prüfer transformation and the implicit function theorem in Banach spaces play key roles in the proofs.