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Re: Anomaly? or at least a surprise.

  • To: mathgroup at
  • Subject: [mg90284] Re: Anomaly? or at least a surprise.
  • From: "David Park" <djmpark at>
  • Date: Fri, 4 Jul 2008 03:59:26 -0400 (EDT)
  • References: <g4i942$3cr$>

First notice that

D[ArcTan[x]]         gives


Plot[D[ArcTan[x]], {x, -10, 10}, PlotRange -> Full]

is just a plot of ArcTan[x], if that is what you wanted. You might as well 
just have used ArcTan[x]. But

Plot[D[ArcTan[x], x], {x, -10, 10}, PlotRange -> Full]

is evaluated by substituting numerical values of x into the plot expression. 
For example, when Mathematica is evaluating for x = .5, say, it tries to 

D[ArcTan[.5], .5]

which is nonsense because the second argument is not a symbolic variable 
and, even if it was, the first argument does not contain a symbolic variable 
but is evaluated to a number.

But the plot statement will work if you Evaluate the plotting expression.

Plot[D[ArcTan[x], x] // Evaluate, {x, -10, 10}, PlotRange -> Full]

Which all leads to a useful, but certainly not ironclad, rule. Don't cram a 
lot of calculation into plotting expressions. Look at them first, outside 
the plotting statement, and make certain you know what they are. Then you 
might even define a function for them and put that into the plotting 

f[x_] = D[ArcTan[x], x]

Plot[f[x], {x, -10, 10}, PlotRange -> Full]

Or, in this case, there is even a simpler solution:

Plot[ArcTan'[x], {x, -10, 10}, PlotRange -> Full]

David Park
djmpark at

"Steve Gray" <stevebg at> wrote in message 
news:g4i942$3cr$1 at
> Doing D[ArcTan[x], x] gives 1/(x^2+1) as expected.
> Doing Plot[D[ArcTan[x]],{x,-10,10},PlotRange->Full] with no variable
> to differentiate by gives a correct plot (!) This is surprising - Help
> says nothing about leaving out the independent variable even when the
> function is of only one variable. Now, doing
> Plot[D[ArcTan[x], x], {x, -10, 10}, PlotRange -> Full]
> gives error messages such as
> General::ivar: -9.18326 is not a valid variable. >>
> General::ivar: -9.59143 is not a valid variable. >>, etc.
> I thought that D[ArcTan[x], x] and 1/(x^2+1) should behave identically
> in the Plot statement.
> Is this behavior something I should know? Thank you.
> Steve Gray

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