Why doesn't NDSolve StepMonitor return values for derivative term?

Questions
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1

In the evaluation below, I want both y[t] and y’[t] sown by StepMonitor, but for some reason y[t] is sown unevaluated. It appears from the output that the derivative is taken after t is replaced by its value like y’[t->value], rather than y’[t]/.t->value. Does anyone see my error?

{sol, steps} = 
  Reap[NDSolveValue[{y'[t] == -y[t], y[0] == 1}, {y[t], y'[t]}, {t, 0,
      5}, StepMonitor :> Sow[{t, y[t], y'[t]}]]];

Plot[sol, {t, 0, 5}, PlotLegends -> {"y[t]", "y'[t]"}]

StepMonitor.png
StepMonitor.png427×214 2.85 KB

steps[[1, 1 ;; 5]]

{{0.000190613, 0.999809, Derivative[1][y][0.000190613]}, {0.000381227,
   0.999619, Derivative[1][y][0.000381227]}, {0.00489873, 0.995113, 
  Derivative[1][y][0.00489873]}, {0.00941624, 0.990628, 
  Derivative[1][y][0.00941624]}, {0.0139337, 0.986163, 
  Derivative[1][y][0.0139337]}}
2

At the time when StepMonitor is invoked, there is no solution available to differentiate yet. There is just a discrete grid of points t_1, t_2, \ldots for each time step and corresponding approximate solution values y_1, y_2, \ldots . The plot works because the solution has already been built up as an interpolating function, and Derivative knows what to do with these.

Of course, if solving an equation of the form y'(t) = f(t, y), then why not just use Sow[f[t, y[t]] to effectively sow the derivative values at each step?

3

Thank you, Ilian. That makes perfect sense. Now, please pour yourself a nice beer and have a very Merry Christmas!

Warm regards, David