A spectrum that is the result of applying a derivative transform to the data of the original spectrum. Derivatives of spectra are very useful for two reasons:

1. First, and second derivatives may swing with greater amplitude than the primary spectra. For example, a spectrum suddenly changes from a positive slope to a negative slope, such as at the peak of a narrow feature (see the figure below). The more distinguishable derivatives are especially useful for separating out peaks of overlapping bands.

2. In some cases derivative spectra can be a good noise filter since changes in base line have negligible effect on derivatives. For example, scattering increases with wavelength for some biologically active macromolecules causing an increasing slope of the absorbance baseline.

A commonly used approximation of the first derivative is: dα/dλ = [α (λ + Δλ) - α (λ - Δλ)] / 2Δλ.

A more accurate approximation of the first and higher order derivatives is presented in thorough explanations by Whitaker¹ and Morrey². Still other methods involve a best fit match to the curve on the features of interest and performing higher order derivatives with numerical analysis.

Derivative spectra yield good signal-to-noise ratios only if the difference of noise levels at the endpoints of the interval is small enough to yield a noise equivalent Δdα/dλ calculation much smaller than the absorbance.

¹Stephen Whitaker and R. L. Pigford, "Numerical Differentiation of Experimental Data", Industrial and Engineering Chemistry, vol. 52, no. 2 February 1960, pp.185 - 187.

²J. R. Morrey, "On Determining Spectral Peak Positions from Composit Spectra with a Digital Computer", Analytical Chemistry, vol. 40, no. 6, May 1968, pp. 905 - 914.