For our first example, we will differentiate data, obtained (for the convenience of comparison) from a polynomial source. First, generate the times and signal Vectors, and note the time step, all using the GenerateSignal command:

$f≔-t^2+7t+4$

$f≔-t^2+7t+4$

$a≔0\:$

$b≔5\:$

$n≔250\:$

$T,\mathrm{dt},X≔\mathrm{GenerateSignal}\left(f\,t=a..b\,n\,'\mathrm{output}'=\left['\mathrm{times}'\,'\mathrm{timestep}'\,'\mathrm{signal}'\right]\right)$

Now, differentiate:

$\mathrm{DX}≔\mathrm{DifferentiateData}\left(X\,1\,'\mathrm{step}'=\mathrm{dt}\,'\mathrm{method}'='\mathrm{forward}'\,'\mathrm{extrapolation}'='\mathrm{polynomial}'\right)$

Since the data does not have any erratic behavior, we expect this approximation to be good:

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{DX}\,\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\,t\right)\,t=a..b\,n\right)\right)$

$0.00570236649300809566$

Visually:

$\mathrm{dataplot}\left(T\,\left[X\,\mathrm{DX}\right]\,'\mathrm{style}'='\mathrm{line}'\,'\mathrm{color}'=\left['\mathrm{blue}'\,'\mathrm{darkgreen}'\right]\,'\mathrm{legend}'=\left[''Signal''\,''First\; derivative''\right]\right)$

In this example, we consider real-valued, periodic data. First, generate a signal, along with the times Vector and time step:

$f≔5\cos \left(t\right)+\sin \left(3t\right)$

$f≔5\cos \left(t\right)+\sin \left(3t\right)$

$a≔0\:$

$b≔2\mathrm{\pi }\:$

$n≔64\:$

$T,\mathrm{dt},X≔\mathrm{GenerateSignal}\left(f\,t=a..b\,n\,'\mathrm{includefinishtime}'='\mathrm{false}'\,'\mathrm{output}'=\left['\mathrm{times}'\,'\mathrm{timestep}'\,'\mathrm{signal}'\right]\right)\:$

Second, compute the first derivative three ways, one directly with the GenerateSignal command (for comparison), and two with the DifferentiateData using different options:

$Y≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\,t\right)\,t=a..b\,n\,'\mathrm{includefinishtime}'='\mathrm{false}'\right)\:$

$\mathrm{Y1}≔\mathrm{DifferentiateData}\left(X\,1\,'\mathrm{step}'=\mathrm{dt}\,'\mathrm{method}'='\mathrm{central}'\,'\mathrm{extrapolation}'='\mathrm{periodic}'\right)\:$

$\mathrm{Y2}≔\mathrm{DifferentiateData}\left(X\,1\,'\mathrm{step}'=\mathrm{dt}\,'\mathrm{method}'='\mathrm{spectral}'\right)\:$

The central approximation for the first derivative is fairly accurate, and the spectral approximation is exceptionally accurate:

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{Y1}\,Y\right)$

$0.00753297720724038257$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{Y2}\,Y\right)$

$4.77375076333675125\times {10}^{\mathrm{-15}}$

Third, compute the second derivative, again in three ways, and calculate the accuracy of the approximations:

$Z≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\,\mathrm{`\$`}\left(t\,2\right)\right)\,t=a..b\,n\,'\mathrm{includefinishtime}'='\mathrm{false}'\right)\:$

$\mathrm{Z1}≔\mathrm{DifferentiateData}\left(X\,2\,'\mathrm{step}'=\mathrm{dt}\,'\mathrm{method}'='\mathrm{central}'\,'\mathrm{extrapolation}'='\mathrm{periodic}'\right)\:$

$\mathrm{Z2}≔\mathrm{DifferentiateData}\left(X\,2\,'\mathrm{step}'=\mathrm{dt}\,'\mathrm{method}'='\mathrm{spectral}'\right)\:$

As with the first derivative, the approximations for the second derivative are fairly accurate (but less so than for the first derivative):

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{Z1}\,Z\right)$

$0.0250341382145791123$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{Z2}\,Z\right)$

$6.56329427489268421\times {10}^{\mathrm{-14}}$

Visually:

$\mathrm{dataplot}\left(T\,\left[X\,Y\,Z\right]\,'\mathrm{style}'='\mathrm{line}'\,'\mathrm{color}'=\left['\mathrm{blue}'\,'\mathrm{darkgreen}'\,'\mathrm{purple}'\right]\,'\mathrm{legend}'=\left[''Signal''\,''First\; derivative''\,''Second\; derivative''\right]\right)$

When we constructed the data, we omitted the point at $t=2\mathrm{\pi }$ since it corresponds to the point at $t=0$. If the final point is included, the spectral approximation for the derivative is greatly affected at the endpoints:

$U,h,V≔\mathrm{GenerateSignal}\left(f\,t=a..b\,n\,'\mathrm{includefinishtime}'='\mathrm{true}'\,'\mathrm{output}'=\left['\mathrm{times}'\,'\mathrm{timestep}'\,'\mathrm{signal}'\right]\right)\:$

$\mathrm{W1}≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\,t\right)\,t=a..b\,n\,'\mathrm{includefinishtime}'='\mathrm{true}'\right)\:$

$\mathrm{W2}≔\mathrm{DifferentiateData}\left(V\,1\,'\mathrm{step}'=h\,'\mathrm{method}'='\mathrm{spectral}'\right)\:$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{W2}\,\mathrm{W1}\right)$

$0.106321866987496502$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{W2}\left[20..-20\right]\,\mathrm{W1}\left[20..-20\right]\right)$

$0.00768251003253147581$

First, generate a signal:

$f≔\mathrm{piecewise}\left(t<\mathrm{\pi }\mathbf{or}3\mathrm{\pi }<t\&comma;\sin \left(4t\right)+5\&comma;\frac{1}{10}\sin \left(40t\right)+5\right)$

$f≔\left\{\begin{array}{cc}\sin \left(4t\right)+5& t<\mathrm{\pi }\mathbf{or}3\mathrm{\pi }<t\\ \frac{\sin \left(40t\right)}{10}+5& \mathrm{otherwise}\end{array}\right.$

$a≔0\&colon;$

$b≔4\mathrm{\pi }\&colon;$

$n≔2^{14}\&colon;$

$X,\mathrm{dt}≔\mathrm{GenerateSignal}\left(f\&comma;t=a..b\&comma;n\&comma;'\mathrm{includefinishtime}'='\mathrm{false}'\&comma;'\mathrm{output}'=\left['\mathrm{signal}'\&comma;'\mathrm{timestep}'\right]\right)\&colon;$

Second, compute the first and second derivatives, both directly and numerically:

$\mathrm{D1X1}≔\mathrm{DifferentiateData}\left(X\&comma;1\&comma;'\mathrm{step}'=\mathrm{dt}\&comma;'\mathrm{method}'='\mathrm{spectral}'\right)\&colon;$

$\mathrm{D2X1}≔\mathrm{DifferentiateData}\left(X\&comma;2\&comma;'\mathrm{step}'=\mathrm{dt}\&comma;'\mathrm{method}'='\mathrm{spectral}'\right)\&colon;$

$\mathrm{D1X2}≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\&comma;t\right)\&comma;t=a..b\&comma;n\&comma;'\mathrm{includefinishtime}'='\mathrm{false}'\right)\&colon;$

$\mathrm{D2X2}≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\&comma;t\&comma;t\right)\&comma;t=a..b\&comma;n\&comma;'\mathrm{includefinishtime}'='\mathrm{false}'\right)\&colon;$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{D1X1}\&comma;\mathrm{D1X2}\right)$

$3.52534051901222011\times {10}^{\mathrm{-7}}$

$\mathrm{RelativeRootMeanSquareError}\left(\mathrm{D2X1}\&comma;\mathrm{D2X2}\right)$

$0.0000559512708099739738$

Finally, plot the first and second derivatives:

$\mathrm{SignalPlot}\left(X\&comma;'\mathrm{title}'=''Plot\; of\; Signal''\&comma;'\mathrm{color}'='\mathrm{blue}'\right)$

$\mathrm{SignalPlot}\left(\mathrm{D1X1}\&comma;'\mathrm{title}'=''Plot\; of\; First\; Derivative''\&comma;'\mathrm{color}'='\mathrm{blue}'\right)$

$\mathrm{SignalPlot}\left(\mathrm{D2X1}\&comma;'\mathrm{title}'=''Plot\; of\; Second\; Derivative''\&comma;'\mathrm{color}'='\mathrm{blue}'\right)$

Complex data can also be differentiated:

$T,\mathrm{dt},X≔\mathrm{GenerateSignal}\left(t^2\exp \left(-\mathrm{I}t\right)\&comma;t=0..5\&comma;{10}^4\&comma;'\mathrm{output}'=\left['\mathrm{times}'\&comma;'\mathrm{timestep}'\&comma;'\mathrm{signal}'\right]\right)\&colon;$

$\mathrm{DX}≔\mathrm{DifferentiateData}\left(X\&comma;1\&comma;'\mathrm{step}'=\mathrm{dt}\&comma;'\mathrm{method}'='\mathrm{forward}'\&comma;'\mathrm{extrapolation}'='\mathrm{spline}'\right)\&colon;$

$\mathrm{X1},\mathrm{X2}≔\mathrm{ComplexToReal}\left(X\right)\&colon;$

$\mathrm{DX1},\mathrm{DX2}≔\mathrm{ComplexToReal}\left(\mathrm{DX}\right)\&colon;$

$\mathrm{dataplot}\left(T\&comma;\left[\mathrm{X1}\&comma;\mathrm{DX1}\right]\&comma;'\mathrm{style}'='\mathrm{line}'\&comma;'\mathrm{title}'=''Signal''\&comma;'\mathrm{legend}'=\left[''Real\; part''\&comma;''Imaginary\; part''\right]\&comma;'\mathrm{color}'=\left['\mathrm{red}'\&comma;'\mathrm{blue}'\right]\right)$

$\mathrm{dataplot}\left(T\&comma;\left[\mathrm{X2}\&comma;\mathrm{DX2}\right]\&comma;'\mathrm{style}'='\mathrm{line}'\&comma;'\mathrm{title}'=''First\; Derivative''\&comma;'\mathrm{legend}'=\left[''Real\; part''\&comma;''Imaginary\; part''\right]\&comma;'\mathrm{color}'=\left['\mathrm{red}'\&comma;'\mathrm{blue}'\right]\right)$

The Savitzky-Golay method can also be used to estimate the derivative:

$f≔\sin \left(t\right)+5\exp \left(\mathrm{I}t\right)\cos \left(3t\right)+10$

$f≔\sin \left(t\right)+5{\&ExponentialE;}^{\mathrm{I}t}\cos \left(3t\right)+10$

$a≔0$

$a≔0$

$b≔2\mathrm{\pi }$

$b≔2\mathrm{\pi }$

$n≔250$

$n≔250$

$\mathrm{dt},X≔\mathrm{GenerateSignal}\left(f\&comma;t=a..b\&comma;n\&comma;'\mathrm{includefinishtime}'='\mathrm{false}'\&comma;'\mathrm{output}'=\left['\mathrm{timestep}'\&comma;'\mathrm{signal}'\right]\right)\&colon;$

$Y≔\mathrm{DifferentiateData}\left(X\&comma;1\&comma;'\mathrm{step}'=\mathrm{dt}\&comma;'\mathrm{method}'='\mathrm{savitzkygolay}'\&comma;'\mathrm{frameradius}'=10\&comma;'\mathrm{extrapolation}'='\mathrm{periodic}'\&comma;'\mathrm{degree}'=3\right)\&colon;$

Comparing with the actual derivative:

$Z≔\mathrm{GenerateSignal}\left(\mathrm{diff}\left(f\&comma;t\right)\&comma;t=a..b\&comma;n\&comma;'\mathrm{includefinishtime}'='\mathrm{false}'\right)\&colon;$

$\mathrm{RelativeRootMeanSquareError}\left(Y\&comma;Z\right)$

$0.00205478287629022073$