Differentiate and Integrate Interpolated Data with Quantities 

Perform three-dimensional interpolation on a list of quantity data.

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X data = N@Flatten[ Outer[Quantity[Sin[#1 + #2^2], "Pounds"] &, Range[10], Range[10]], 1]; grid = Flatten[ Outer[List, Quantity[Range[10], "Seconds"], Quantity[Range[10], "Feet"]], 1]; data = Join[grid, Transpose[{data}], 2] /. {x_, y_, z_} :> {{x, y}, z};

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X if = Interpolation[data]

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Plot the solution.

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X fun[t_?NumericQ, p_?NumericQ] := QuantityMagnitude[ Quiet[if[Quantity[t, "Seconds"], Quantity[p, "Feet"]] ] ]; Plot3D[fun[tt, p], {tt, 1, 10}, {p, 1, 10}, AxesLabel -> {"Seconds", "Feet", "Pounds"}, PlotRange -> All, ImageSize -> Medium]

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The input and output quantities associated with the interpolated function can be extracted using property "QuantityUnits".

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X if["QuantityUnits"]

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Differentiate the interpolated function.

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X Derivative[#, 0][if][Quantity[3, "Seconds"], Quantity[1, "Meters"]] & /@ Range[3]

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X Derivative[#, #][if][Quantity[3, "Seconds"], Quantity[1, "Meters"]] & /@ Range[3]

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Integrate the interpolated function.

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X Integrate[ if[x, y], {x, Quantity[1, "Seconds"], Quantity[10, "Seconds"]}, {y, Quantity[1, "Feet"], Quantity[10, "Feet"]}]

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