Clear["Global`*"]
f = 2 a ArcTanh[(# a)/Sqrt[-1 + #^2 b]] + a Log[1 + #^2 (a^2 - b)] -
2 Log[# b + Sqrt[-1 + #^2 b]] & /. {a -> 2/10, b -> 3/10};
Plot[Abs[f[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f(x)|"}]

Use ParametricPlot
to plot the inverse
ParametricPlot[{Abs[f[x]], x}, {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"|f(x)|", "x"}]

Use "Log" scaling (ScalingFunctions
) for the derivative
Plot[Abs[f'[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f'(x)|"},
ScalingFunctions -> "Log"]

as well as for the ratio
Plot[Abs[f[x]/f'[x]], {x, -5, 5},
PlotRange -> All,
PlotPoints -> 100,
MaxRecursion -> 10,
AspectRatio -> 1,
AxesLabel -> {"x", "|f(x)/f'(x)|"},
ScalingFunctions -> "Log"]

f[x]
is complex, is this feature intended? If yes your question cann#t be answered. $\endgroup$ – Ulrich Neumann Dec 30 '20 at 13:32Re, Im, Abs,...
$\endgroup$ – Ulrich Neumann Dec 30 '20 at 13:47