Plot[ D[Sin[x]] ] and Evaluate[] [duplicate]

Question

Why does

t = 2 Pi;
Plot[D[Sin[x],x], {x,0,t}] (* Plotting the derivative of Sin[x] *)

not work, but

t = 2 Pi;
Plot[Evaluate[D[Sin[x],x], {x,0,t}] (* Plotting the evaluation of the derivative of Sin[x]? *)

do? And why does this work, but neither

t = 2 Pi;
Plot[{D[Sin[x],x]}, {x,0,t}] (* Plotting the one-length array of the derivative of Sin[x] *)

nor

t = 2 Pi;
Plot[{Evaluate[D[Sin[x],x]]}, {x,0,t}] (* Plotting the one-length array of the evaluation of the derivative of Sin[x]? *)

works?

Answer 1

I always picture it like this: Plot has attribute HoldAll, so it gets the unevaluated expression D[Sin[x],x]. Then it replaces all occurrences of x with 5 and evaluates the result. So it tries to evaluate something like D[Sin[5],5]. Which is of course nonsensical, because you can't derive by a constant.

If you call Plot[Evaluate[D[Sin[x],x], ..., the expression gets evaluated before it's passed to Plot. So it's equivalent to Plot[Cos[x], {x, 0, t}]. Now if Plot replaces x by 5, it gets a nice real number it can plot.

However, this only happens when Evaluate is the head of the expression passed to Plot. If you pass something like x+5/(Evaluate[y]), the subexpression y isn't evaluated. HoldAll doesn't "look inside" the expression, to see if there's an Evaluate nested somewhere. That's why Plot[{Evaluate[D[Sin[x],x]]}, ... doesn't work.

However, Plot[Evaluate[{D[Sin[x], x]}], {x, 0, t}] does work, because Evaluate is the head of the expression