Why doesn't plotting a derivative work?

Question

If I plot a derivative it fails: Plot[ D[x^2 + 1, x], {x, -6., 6.}] General::ivar: -5.99975 is not a valid variable.

But if I simply put the derivative, 2x, in, the plot works: Plot[2x,{x,-.6,.6}]

Most other standard functions don't do this. Why doesn't plotting a derivative work, and what is the "..valid variable" warning about?

There is another question just a bit before about the same question. That is how popular it is. Did you check the search box here for "why plot derivative", for example ? You can get answers a lot faster and with less effort by searching. Searching what I mentioned before I found Can't plot derivative of Hankel function. Since this question is very popular it already had a duplicate — userrandrand, Oct 09 '22 at 22:40
Oddly enough, I discovered using ' instead of D does work, which that article does not answer. f[x_] := x^2 + 1 Plot[f'[x], {x, -5, 5}] --Works Plot[D[f[x], x], {x, -5, 5}] --Fails — cybervigilante, Oct 09 '22 at 23:49
I got rid of set delayed after reading the article but got the same result D fails, and ' works, which makes no sense to me. The Help on Hold is really mysterious. Doesn't say much. — cybervigilante, Oct 10 '22 at 00:01
Hi, that answer does explain why f'[x] works when D[f[x],x] does not. During the plot x is replaced directly by one of the values in the range specified. For example, starting at x=-5 it will evaluate D[f[-5],-5] which is not valid as you can't take the derivative of a number. However f'[-5] is fine whether you evaluate it your self or Plot evaluates it. f'[x] and D[f[x],x] have the same output but they are encoded differently. To see that you can use //HoldForm//FullForm on both. — userrandrand, Oct 10 '22 at 01:17
In particular, you can evaluate f' without choosing an argument and it will output the derivative as a function (you can try with f[x_]:=x^3). Hence basically, f'[4] is basically the same as g=f'; g[4] and there is no problem in plotting g[x]. — userrandrand, Oct 10 '22 at 01:19
Does this answer your question? General::ivar is not a valid variable when plotting - what actually causes this and how to avoid it? — xzczd, Oct 10 '22 at 05:31

score 2 · Answer 1 · answered Oct 10 '22 at 01:22

2

Using With:

With[{f = D[x^2 + 1, x]}, Plot[f, {x, -6, 6}]]

Or using HoldForm as points out @userrandrand:

ReleaseHold@HoldForm[Plot][D[x^2 + 1, x], {x, -6, 6}]

answered Oct 10 '22 at 01:22

E. Chan-López

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Hi. I did not mention using HoldForm like that but that's a possibility. Although the idea of controlling the evaluation order is similar, I think it's quicker to use Plot[D[x^2+1,x]//Evaluate,{x,-6,6}]. Another possibility is D[x^2 + 1, x] // Plot[#, {x, -6, 6}] &. That last option is particularly convenient if there a lot of functions that will be plotted in the same interval in which case one can set plt=Plot[#, {x, -6, 6}] & and use D[x^2 + 1, x] //plt then D[x^4 + 1, x] //plt – userrandrand Oct 10 '22 at 01:37
Hi. True, you didn't mention using HoldForm in that way, but I thought you were thinking of that. In fact, D[x^2 + 1, x] // Plot[#, {x, -6, 6}] & is a better way. :-) – E. Chan-López Oct 10 '22 at 01:40
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Another way is plt2 = OperatorApplied[Plot] then D[x^2 + 1, x] // plt2[{x, -6, 6}] which could be convenient for changing the interval. – userrandrand Oct 10 '22 at 01:53
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@userrandrand - or more concisely, Plot @@ {D[x^2 + 1, x], {x, -6, 6}} – Bob Hanlon Oct 10 '22 at 02:36
@BobHanlon that one is quite beautiful. Feels like something I should save in a corner of my log on screen . Quite pretty. – userrandrand Oct 10 '22 at 02:42

Why doesn't plotting a derivative work?

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