Gianluca, I got the impression that the original poster wants the curves to be labeld by the functions itself. Not just "function" and "derivative". So this a simple variation of your solution:

var = Input["Type a variable, such as x, in this window."];
expr = Input[
   "Type an expression to be differentiated in the variable you just typed in the last window."];
g[blah_] = expr /. {var -> blah};
derivatives := {Callout[g[var], g[var], Above], Callout[g'[var], g'[var], Above]};
Plot[
    Evaluate@derivatives, {x, -3, 5},
    PlotLabel -> "A function and its Derivative", PlotRange -> 20
 ]

I increased the plot range, so as not to crowd the function labels

Callout documentation

Ok good news. I figured out PlotRange.. I changed it to:

PlotRange -> {-10,10}

I knew it was something easy..Now what about these damn labels on the actual curves??

I do see I use "PlotLabels" twice...For my actual title of the plot output, should I be using a different command to label the cur