Function Repository Resource:

PiecewiseD (1.0.0) current version: 2.0.1 »

The derivative of a piecewise function with Indeterminate for points or regions where the function is not defined

Contributed by: Dennis M Schneider

ResourceFunction["PiecewiseD"][f,x]

returns the derivative of a piecewise function returning the value Indeterminate for points or regions where the function is not defined.

ResourceFunction["PiecewiseD"][f,x,k]

returns the function together with its first k derivatives.

ResourceFunction["PiecewiseD"][f,{x,k}]

returns the k^th derivative.

Examples

Basic Examples (3)

Compute the derivative of a piecewise function:

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$f[x_] := \!$\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"-", "2"}], " ", "x"}], RowBox[{"x", "<", "0"}]}, { RowBox[{ RowBox[{"2", " ", "x"}], "+", "1"}], RowBox[{"x", ">", "0"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}, Selectable->True]} }, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{1.}}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False, StripWrapperBoxes->True]$ ResourceFunction["PiecewiseD"][f[x], x]$

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Plot a function together with its piecewise derivative:

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Compute the first two derivatives of a function whose domain is not an interval:

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$Clear[f] f[x_] := \!\(\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{"x", "-", RowBox[{"1", "/", "2"}]}], RowBox[{"x", "<", "0"}]}, { RowBox[{"Sin", "[", "x", "]"}], RowBox[{"x", ">=", "1"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False, StripWrapperBoxes->True]\) ResourceFunction["PiecewiseD"][f[x], x, 2]$

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Compute just the second derivative:

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Plot the function together with its first two derivatives:

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Scope (4)

Find and plot the first- and second-order derivatives. The function and its first-order derivative are continuous at x=0, but not the second-order derivative:

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Check that the first derivative is continuous:

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Check that the second derivative is not continuous:

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Plot the results:

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$ResourceFunction["EnhancedPlot"][ ResourceFunction["PiecewiseD"][f[x], x, 2], {x, -\[Pi], \[Pi]}, "Exception" -> 0, PlotRange -> {-3.5, 3.5}]$

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Properties and Relations (1)

Show the difference between PiecewiseD and D:

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Applications (4)

The following function has a removable discontinuity at x=2 and an infinite discontinuity at x=4:

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$f[x_] := (x^2 - 4)/((x - 2) (x - 4)) ResourceFunction["EnhancedPlot"][f[x], {x, 0, 6}, "Exception" -> 2, "Asymptote" -> 4]$

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Extend the definition at x=2 to make the extended function continuous there:

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$g[x_] := \!\(\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{"f", "[", "x", "]"}], RowBox[{ RowBox[{"x", "!=", "2"}], "&&", RowBox[{"x", "!=", "4"}]}]}, { RowBox[{"-", "2"}], RowBox[{"x", "==", "2"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False, StripWrapperBoxes->True]\) ResourceFunction["EnhancedPlot"][g[x], {x, 0, 6}, "Exception" -> 2, "Asymptote" -> 4]$

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The extended function is actually differentiable at x=2:

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The resource function EnhancedPlot produces a correct plot:

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The function g is differentiable at x=0 and PiecewiseD returns the correct value, 1. The function D, however, returns the value 0 for the derivative at x=0:

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$g[x_] := \!\(\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{"4", SuperscriptBox["x", "2"], RowBox[{"Sin", "[", FractionBox["1", "x"], "]"}]}], "+", "x"}], RowBox[{"x", "!=", "0"}]}, {"0", RowBox[{"x", "==", "0"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False]\) {ResourceFunction["PiecewiseD"][g[x], x], D[g[x], x]}$

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However, the derivative is not continuous:

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This function is differentiable at x=0 and its derivative is continuous there:

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$Clear[g] g[x_] := \!\(\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ RowBox[{ SuperscriptBox["x", "3"], RowBox[{"Sin", "[", FractionBox["1", "x"], "]"}]}], "+", "x"}], RowBox[{"x", "!=", "0"}]}, {"0", RowBox[{"x", "==", "0"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False]\) dg[x_] = ResourceFunction["PiecewiseD"][g[x], x]$

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Plot the result using the resource function EnhancedPlot:

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ResourceFunction["EnhancedPlot"][
ResourceFunction["PiecewiseD"][g[x], x], {x, -.5, .5}, "Exception" -> 0, "PointStyle" -> AbsolutePointSize[10]]

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A function with a singularity at x=-1 and x=1; PiecewiseD returns the correct result. Note that if this expression is simplified, the singularity at x=1 will be lost:

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${ResourceFunction["PiecewiseD", ResourceVersion->"1.0.0"][(x^2 + 2 x - 3)/(x^2 - 1) Sin[\[Pi] x], x], Simplify[ResourceFunction["PiecewiseD", ResourceVersion->"1.0.0"][(x^2 + 2 x - 3)/(x^2 - 1) Sin[\[Pi] x], x]]}$

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The resource function EnhancedPlot is able to produce a correct plot:

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$ResourceFunction["EnhancedPlot"][ ResourceFunction["PiecewiseD", ResourceVersion->"1.0.0"][(x^2 + 2 x - 3)/(x^2 - 1) Sin[\[Pi] x], x, 1], {x, -2, 2}, "Exception" -> {-1, 1}, PlotRange -> {-10, 12}]$

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Extend the function so that it becomes continuous at -1 and 1:

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$Limit[(x^2 + 2 x - 3)/(x^2 - 1) Sin[\[Pi] x], x -> #] & /@ {-1, 1}$

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$Clear[h] h[x_] := \!\(\* TagBox[GridBox[{ {"\[Piecewise]", GridBox[{ { RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["x", "2"], "+", RowBox[{"2", "x"}], "-", "3"}], RowBox[{ SuperscriptBox["x", "2"], "-", "1"}]], RowBox[{"Sin", "[", RowBox[{"\[Pi]", " ", "x"}], "]"}]}], RowBox[{ RowBox[{"x", "!=", RowBox[{"-", "1"}]}], "&&", RowBox[{"x", "!=", "1"}]}]}, { RowBox[{ RowBox[{"-", "2"}], "\[Pi]"}], RowBox[{"x", "==", RowBox[{"-", "1"}]}]}, {"0", RowBox[{"x", "==", "1"}]} }, AllowedDimensions->{2, Automatic}, Editable->True, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.84]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}, Selectable->True]} }, GridBoxAlignment->{ "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, "RowsIndexed" -> {}}, GridBoxItemSize->{ "Columns" -> {{Automatic}}, "ColumnsIndexed" -> {}, "Rows" -> {{1.}}, "RowsIndexed" -> {}}, GridBoxSpacings->{"Columns" -> { Offset[0.27999999999999997`], { Offset[0.35]}, Offset[0.27999999999999997`]}, "ColumnsIndexed" -> {}, "Rows" -> { Offset[0.2], { Offset[0.4]}, Offset[0.2]}, "RowsIndexed" -> {}}], "Piecewise", DeleteWithContents->True, Editable->False, SelectWithContents->True, Selectable->False, StripWrapperBoxes->True]\)$

The first and second derivatives are continuous at ±1:

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