f[x_] = x/(x^2 + 1)  fp[x_] = Simplify[D[f[x], x]]fpp[x_] = Simplify[D[fp[x], x]]

D[f[x], x]D[fp[x], x]

Plot[{f[x], fp[x], fpp[x]}, {x, -2, 2}, 
      PlotStyle -> {Hue[0.5], Hue[0.6], Hue[0.7]}]

xs  =  {0,  5,  10,  15,  20}slopes = {20/10,   100/10,  400/5, 500/10,  500/10}

plt=ListPlot[Transpose[{xs, slopes}],PlotStyle -> PointSize[.02], PlotJoined -> True]

plt=ListPlot[Transpose[{xs, slopes}],PlotStyle -> PointSize[.02], PlotJoined -> False]

<< Graphics`Spline`

\(\(		
    Show[plt, 
\ \ 
      Graphics[

        		{Hue[0], \ Line[pts], \ GrayLevel[0], 
\ \ \ \ 
          Spline[Transpose[{xs, slopes}], \ Cubic]}
		], 
\ \ \ \ 
      PlotRange\  -> \ All]\)\)

secondslopes = {30/20,   70/15, 0, 0,  0}ListPlot[Transpose[{xs, secondslopes}],PlotStyle -> PointSize[.02], PlotJoined -> True]