sine_derivative_animation.nb

x0s = Table[x0, {x0, 0, 2 \[Pi], 2 \[Pi]/10}];

x0labs = Table[x0, {x0, 0, 2 \[Pi], \[Pi]/4}];

Clear[y]
y[f_, x0_] := ( D[f[x], x] /. x -> x0)*(x - x0) + f[x0] 

Clear[func]
func[x0_, \[CapitalDelta]y_: 0, \[CurlyEpsilon]_: 1] := Module[{k},
  
  k = y[Sin, x0] + \[CapitalDelta]y;
  
  Plot[ k, {x, x0 - \[CurlyEpsilon], x0 + \[CurlyEpsilon]}, 
   PlotStyle -> { Opacity[.9], Green}, Background -> Black, 
   AxesStyle -> White, Ticks -> {x0labs, {-1, 1}}]
  
  ]
  
  Clear[makeline]


makeline[x0_, \[CapitalDelta]y_: 0, \[CurlyEpsilon]_: 1, dot_: False, 
   i_: 1, op_: 1] := Module[{frames},
   
   If[i == 1, frames = {Nothing}, frames = finalFrames[[;; i - 1]]];
   
   If[dot == False,
    
    Show[ 
     Plot[Sin[x], {x, -\[Pi], 3 \[Pi]}, 
      PlotRange -> {{-.6, 7}, {-1.5, 1.5}}], frames, 
     func[x0, \[CapitalDelta]y, \[CurlyEpsilon]], 
     Background -> Black, AxesStyle -> White, 
     Ticks -> {x0labs, {-1, 1}}],
    
    Show[ 
     Plot[Sin[x], {x, -\[Pi], 3 \[Pi]}, 
      PlotRange -> {{-0.6, 7}, {-1.5, 1.5}}, 
      PlotStyle -> Opacity[op]], 
     Graphics[{Opacity[op], PointSize[.015], Green, 
       Point[{x0, Cos[x0]}] }], frames, Background -> Black, 
     AxesStyle -> White, Ticks -> {x0labs, {-1, 1}}]
    
    ]
   
   ];
   
   Clear[movetoderiv]

movetoderiv[x0_, i_: 1] := Module[{dot, deriv},
  
  
  Table[
   
   If[\[CapitalDelta]y == Cos[x0] - Sin[x0], dot = True, 
    dot = False];
   
   makeline[x0, \[CapitalDelta]y, 
    E^(-2 Abs[\[CapitalDelta]y]) ((
      Abs[Cos[x0] - Sin[x0]] - Abs[\[CapitalDelta]y])/(
      Abs[Cos[x0] - Sin[x0]] + 10^(-10))), dot , i]
   
   , {\[CapitalDelta]y, 0, Cos[x0] - Sin[x0] , (Cos[x0] - Sin[x0])*.1}]
  ]
  
  Clear[initLine]

initLine[x0_, i_: 1] := 
 Table[makeline[x0, 0, \[CurlyEpsilon], False, 
   i], {\[CurlyEpsilon], .01, 1, .05}]
   
   Clear[animAtX]
   

animAtX[x0_, i_: 1] := Module[{frames},
   
   Join[ Join[ initLine[x0, i] , 
     Table[ initLine[x0, i][[-1]], {k, 0, 1, .2} ]  ]  , 
    movetoderiv[x0, i] ]
   
   ];
   

finalFrames = makeline[#, 0, 0, True, 1] & /@ x0s ;

finals[op_] := makeline[#, 0, 0, True, 1, op] & /@ x0s ;

tabs = Table[ animAtX[x0s[[i]], i] , {i, 1, Length[x0s]}];

end = Table[  
    Show[  finals[1 - upp/(3 \[Pi])], 
     Plot[Cos[x], {x, 0, upp} , 
      PlotRange -> {{-0.6, 7}, {-1.5, 1.5}}, PlotStyle -> Green, 
      Background -> Black, AxesStyle -> White, 
      Ticks -> {x0labs, {-1, 1}}]]
    
    , {upp, 0, 3 \[Pi], .2} ][[2 ;;]]   ;
    
    
Export["test.gif", Join[Flatten[tabs], end], ImageResolution -> 150, 
 "DisplayDurations" -> .05]