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Analytical expressions are obtained for the derivatives of Mie scattering coefficients with respect to the electrical radius of the spherical scattering particle, and to the relative permittivity and permeability of both the particle and the surrounding medium. Their corresponding approximate expressions are developed to avoid numerical overflow based on the logarithmic derivative of Riccati–Bessel functions. The analytical expressions have been verified by comparing their results with those calculated by analytical expressions developed by Mathematica. Compared with the numerical derivative, the analytical expressions and approximate expressions show a higher accuracy and are 2.0 and 2.8 times, respectively, faster in the case of a single magnetodielectric sphere. Generally, for spheres with an electrical radius in a large range, the approximate expressions can yield acceptable accuracy and computation time up to a high order. This work can be used in the design of nonmetallic metamaterials, and in the retrieval of aerosol properties from remote sensing data. An example calculation is given for the design of an optical, all-dielectric, mu-negative metamaterial consisting of a simple cubic array of tellurium nanoparticles.
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