def deriv_v_r_func(*, Z_1, Z_2, Z_4, v_φ, ρ, a_0, dderiv_B_φ, B_θ,
    norm_kepler, X, v_r):
    """
    Compute the derivative of v_r
    """
    return (Z_1 * v_φ ** 2 * ρ * norm_kepler - 2 * Z_2 * Z_4 * a_0 *
        norm_kepler + dderiv_B_φ * a_0 * B_θ * (2 * v_φ * norm_kepler - Z_4 *
        X)) / (v_φ ** 2 * ρ * norm_kepler - Z_4 * a_0 * v_r)
