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We report on a technique for extracting an accurate value of the nonlinear inductance in superconducting transmission lines. This novel technique assesses the frequency dependence of the transmission line's nonlinear response. A wideband nonlinear measurement system was used to simultaneously measure the third-order spurious signals at 2f1 - f2, 2f2 - f1 + f2, 2f2 + f1, 3f1, and 3f2 frequencies. Measurements for different values of the fundamental frequencies f1 and f2 allow us to study the spurious signal generation from 1 to 21 GHz. We demonstrate this technique by measuring several superconducting YBa2Cu3O7-x coplanar waveguide transmission line geometries patterned in a single chip at 80 K. The results show a linear frequency dependence of the nonlinear response, indicating a dominant contribution of the nonlinear inductance over the nonlinear resistance omegaDeltaL(i) Gt DeltaR(i). The experimentally obtained nonlinear inductances are then used to determine device-independent measures of the linearity of the thin-film material in order to provide the foundation for modeling the nonlinear response of specific devices.