I was wondering why I was having difficulty plotting
∂[LegendreP[l, m, Cos[x]], x], where $l\in\mathbb{N}$ and $m\in\{-l,~-l+1,\cdots,l-1,~l\}$ as per usual for this function. I even tried the equivalent Subscript[∂, x]LegendreP[l, m, Cos[x]] (I am aware that doesn't come out neatly on this page, but that's what copying the code gives after pasting it).
Is it a variable type issue and if so, how can I get around this?
You can set m to depend on L, something like this. But when L=0 there is a problem so need to check for it.
ClearAll[L0, m, x];
Manipulate[
expr = D[LegendreP[L0, m, Cos[x]], x];
If[L0 != 0,
Plot[expr, {x, -L0, L0}, Frame -> True, ImageSize -> 400,
FrameLabel -> {{"f(x)", None}, {"x", "D[LegendreP[L0,m,Cos[x]],x]"}},
ImagePadding -> 50,
GridLines -> Automatic,
GridLinesStyle -> LightGray,
PlotStyle -> Red],
Plot[expr, {x, -1, 1}, Frame -> True, ImageSize -> 400,
FrameLabel -> {{"f(x)", None}, {"x",
"D[LegendreP[L0,m,Cos[x]],x]"}}, ImagePadding -> 50]
],
{{L0, 1, "L"}, -10, 10, 1, ImageSize -> Small, Appearance -> "Labeled"},
{{m, 1, "m"}, -L0, L0, 1, ImageSize -> Small, Appearance -> "Labeled"}
]
D[LegendreP[L0, m, Cos[x]], x]$\endgroup$D[LegendreP[l, m, Cos[x]], x]not∂[LegendreP[l, m, Cos[x]], x]The 1st form gives-(1/(-1 + Cos[x]^2))((-1 - l) Cos[x] LegendreP[l, m, Cos[x]] + (1 + l - m) LegendreP[1 + l, m, Cos[x]]) Sin[x]$\endgroup$