Graphical representations of the positions of the zero contours of the second derivative
(a) and the third derivative (b) of the free energy in the Hg-Mn-Te-Se system at the cross section for the existence of solid solutions of the phase diagram
Calibration Data treatment Analytes Principal components [R.sup.2] RMSEE Second derivative
Glucose 6 0.992 0.18 Second derivative
Fructose 6 0.997 0.06 Second derivative
Sucrose 6 0.989 0.22 Second derivative
Maltose 6 0.999 0.04 Validation Data treatment [R.sup.2] RMSEP Second derivative
0.987 0.55 Second derivative
0.996 0.28 Second derivative
0.996 0.38 Second derivative
0.997 0.29 RMSEE: root mean square error of estimation; RMSEP: root mean square error of prediction.
Constandinou, "Feature extraction using first and second derivative
extrema (fsde) for real-time and hardware-efficient spike sorting," Journal of Neuroscience Methods, vol.
Integrals in (21) and (22) as well as expressions for the moments [mathematical expression not reproducible] can be converted to the second derivative
of Green's function using Stieltjes convolution.
Twelve different geometrical descriptors, together with first and second derivatives
, are chosen: three coefficients of the first fundamental form, i.e.
In the studies of carbon materials, employing analysis of KVV spectra, alternative methods have been proposed, such as the method of the first derivative and second derivative
. With regard to the first method, it has been largely used to determine the energy and linewidth of specific orbitals, which are used to characterize the material.
These studies have involved a range of equipment, varying in wavelength range, wavelength dispersion mechanism, and optical geometry employed (reflectance, diffusive reflectance, transmittance, or interactance), in data preprocessing (first derivative, second derivative
, and multiplicative scatter correction, MSC), and in the calibration model development method (principal component regression, PCR; multiple linear regression, MLR; PLSR1; PLS[R.sup.2]; and modified partial linear square, MPLSR).
Chan and Tsai  introduced special explicit TDRK methods by including the second derivative
. It involves only one evaluation of f and a number of evaluations of g per step.
To increase the accuracy further and to solve the stiffness problem in ordinary differential equations (ODEs),  derived second derivative
multistep methods for stiff ODEs.
To eliminate the baseline drift and random noise of spectral signal, three mathematical pretreatments (no mathematical processing, first derivative, and second derivative
) were tested.
Similarly, the second derivative
process [[partial derivative].sup.2]x/[partial derivative][a.sup.2] is obtained by differentiating (14) with respect to a, to obtain
of the Time Response Evaluated at t/T = 1