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# Derivative details

Derivative calculation can be carried out with all 2D and 3D data objects, regardless if they contain equidistant or discrete data points. The order of the derivative is variable and can be adjusted by the user. The algorithm used for calculation of the derivative might also be adjusted.

For the derivative calculation, two different algorithms are available in the software:

The derivative workflow is described in more detail here:

Examples: See also

Open file

Mathematics Tab

Savitzky-Golay Smoothing Algorithm

Undo Function

Audit Trail

## Differential Quotient Derivative Algorithm

The derivative for each data point of the 2D data object is calculated from the average differential quotients of two adjacent data points to the data point of interest. This procedure is applied to all data points of the 2D data object. It is assumed, that data points are available in ascending order regarding to the x-axis of the 2D data object.

The derivative is calculated from both sides of a data point P2 as follows:

There are two adjacent data points P1(x1;y1) and P3(x3;y3) next to the data point P2(x2;y2). The differential quotients must be calculated to obtain the slopes S21 between points P2 and P1 and S32 between the data points P3 and P2 as follows: and The derivative D2 of data point P2 is calculated as average of the slopes S21 and S32:  Caution when applying this algorithm to discrete 2D data objects! The distance between data points is neglected, which might have relevant effects when the derivative is applied to discrete 2D data objects.

## Differential Quotient Derivative Example

The following x,y data set shows a sine function: (Source: LabCognition, Analytical Software GmbH & Co. KG, Leyendeckerstr. 33, 50825 Cologne, Germany)

The derivative calculation using the differential quotient algorithm returns a cosine function as expected: (Source: LabCognition, Analytical Software GmbH & Co. KG, Leyendeckerstr. 33, 50825 Cologne, Germany)

## Savitzky-Golay Derivative

The higher order coefficients a1 to aM of the Savitzky-Golay smoothing algorithm are used for computation of numerical derivatives of 2D data objects. The derivative for each data point of the 2D data object is derived from the convolution gi. It has to be multiplied by m! to obtain the mth order derivative coefficients.  Tip:  For derivative calculations the order M of the polynomials should be equal or greater than 4.

## Savitzky-Golay Derivative Example

The following x,y data set shows a sine function: (Source: LabCognition, Analytical Software GmbH & Co. KG, Leyendeckerstr. 33, 50825 Cologne, Germany)

The derivative calculation using the Savitzky-Golay derviative algorithm returns a cosine function as expected: ## Derivative Parameters

The following derivative parameters can be adjusted:

### Derivative order

This value indicates the order of the derivative function applied to the current 2D data object. An integer value greater than 0 must be entered into this text field.

• 1 = first order

• 2 = second order

...

• n = nth order.

### Smoothing window

The smoothing window parameter is only used with the Savitzky-Golay derivative algorithm. An odd number of data points around each data point of the spectrum will be taken into account for the derivative calculation. For details, please refer to the Savtizky-Golay documentation. The number of data points can be selected from the drop down combo box by clicking the icon at the right side of the parameter field. Tip:  With increasing derivative order the number of window points must be increased accordingly. If the number of window points is too small, a math error message is displayed on calculation.

### Use Savitzky-Golay algorithm

This is a flag indicating, whether the differential quotient algorithm (default) or the Savitzky-Golay algorithm is used for derivative calculation. The flag might be toggled by clicking the icon at the right side of the parameter field and selecting a new value from the list.

• No
Differential quotient algorithm

• Yes
Savitzky-Golay algorithm

## References

K. H. Norris and P. C. Williams, Cereal Chem, 61 #2 (1984), 158

Madden, Anal. Chem., 50 #9 (1978), 1383

Steiner et al., Anal. Chem., 44 #11 (1972), 1906

Savitzky A., and Golay, M.J.E. 1964, Analytical Chemistry, vol. 36, pp. 1627â1639.

Hamming, R.W. 1983, Digital Filters, 2nd ed. (Englewood Cliffs, NJ: Prentice-Hall).

Ziegler, H. 1981, Applied Spectroscopy, vol. 35, pp. 88â92.

Bromba, M.U.A., and Ziegler, H. 1981, Analytical Chemistry, vol. 53, pp. 1583â1586.