Function broadened_cross_sections_with_analytical_derivative_from_base 

pub fn broadened_cross_sections_with_analytical_derivative_from_base(
    energies: &[f64],
    base_xs: &[Vec<f64>],
    resonance_data: &[ResonanceData],
    temperature_k: f64,
    instrument: Option<&InstrumentParams>,
) -> Result<BroadenedXsWithDerivative, TransmissionError>

Compute Doppler-broadened cross-sections and their analytical temperature derivative from precomputed unbroadened cross-sections.

Returns Doppler-only cross-sections and derivatives. Resolution broadening is NOT applied (issue #442: must be applied after Beer-Lambert on total T).

Uses doppler_broaden_with_derivative for exact ∂σ/∂T in a single pass (1× broadening), replacing the 3× FD approach.

Returns BroadenedXsWithDerivative: (sigma_k, dsigma_k_dT).