Spectroscopic curves can be subjected to numerical differentiation.
When the data points in a curve are equidistant from each other the Savitzky–Golay convolution method may be used. The best convolution function to use depends primarily on the signal-to-noise ratio of the data. The first derivative (slope, ) of all single line shapes is zero at the position of maximum height. This is also true of the third derivative; odd derivatives can be used to locate the position of a peak maximum.
The second derivatives, , of both Gaussian and Lorentzian functions have a reduced half-width. This can be used to apparently improve spectral resolution. The diagram shows the second derivative of the black curve in the diagram above it. Whereas the smaller component produces a shoulder in the spectrum, it appears as a separate peak in the 2nd. derivative. Fourth derivatives, , can also be used, when the signal-to-noise ratio in the spectrum is sufficiently high.