Using Dynamic with PlotRange inside Manipulate

Question

I am using Manipulate to plot a function that takes a fair amount of time to be plotted. Once it is done I would like to change the y-range where it is plotted. To avoid that the function is evaluated again I am trying to use Dynamic. Following an example in Wolfram Documentation I constructed something like this:
Manipulate[
    Plot[f[a,x],{x,0,xmax},PlotRange->Dynamic[{0,xmax}]],
    {a,amin,amax},{xmax,0.1,5}
]

However, I get an error:

General::prng: Value of option PlotRange -> {0,0.1} is not All,
Full, Automatic, a positive machine number, or an appropriate list of
range specifications.

Any ideas on how to solve this?

Chris Degnen · Answer

This version recomputes f[a, x] when the y-range is changed
f[a_, x_] := a Sin[x]
{amin, amax} = {1, 2};
Manipulate[Plot[f[a, x], {x, 0, xmax},
  PlotRange -> {{0, xmax}, {ymin = -ymax, ymax}}],
 {a, amin, amax}, {xmax, Pi, 2 Pi}, {ymax, 2, 6}]

However, in this version the y-range is fixed ...
f[a_, x_] := a Sin[x]
{amin, amax} = {1, 2};
ymax = 2;
Manipulate[plot = Plot[f[a, x], {x, 0, xmax},
   PlotRange -> {{0, xmax}, {ymin = -ymax, ymax}}],
 {a, amin, amax}, {xmax, Pi, 2 Pi}]

but it can be rescaled in a further manipulate without recalculation.
Manipulate[plot /.
  (PlotRange -> {i_, {j_, k_}}) :> (PlotRange -> {i, {-m, m}}),
 {m, 2, 4}]

Using Dynamic with PlotRange inside Manipulate

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