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Abs[] help please


I may not understand the limitations of Abs[].

This works just fine:
s[L_]=I/(L+1/L);(* a complex function *)
Plot[Evaluate[D[Abs[s[L]],L]],{L,1,2}]
(*The Abs[] of a complex function, which is real, apparently has a
real derivative and it is plotted.*)


However, If I put another "I" in the function definition, the
derivative of the Abs[] appears to have an imaginary component:
s[L_]=I/(L+I/L);(* has I instead of 1 *)
Plot[Evaluate[D[Abs[s[L]],L]],{L,1,2}]

The error message is as follows
Plot::"plnr": (1 - I/L^2)* REAL "is not a mach. size real number".
For some reason a factor of (1-I/L^2) showed up (REAL is some real
function).

Can anyone give me a clue how I might be able to plot the derivative
of the Abs[] of this type of complex function?






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