Derivative of the solutions by parameter

Question

P0 = 0.1;
K = 1000;
Quiet[sol = Solve[{K*P*L == PL, P0 == P + PL, r*P0 == L + PL}, {P, L,PL}][[1]]];
complex = PL /. sol;
Plot[complex, {r, 0.1, 2}]

How can I make the solution into a function of parameter r in order to get the derivative of the solution, D[complex[r],r]?

Answer 1

You can do it nicely by defining a function complex

Clear[complex];
complex[x_] := PL /. sol /. r -> x;
Plot[{complex[r], Hold[Evaluate@D[complex[r], r]]} // Release, {r,0.1, 2}]

Otherwise as mentioned in the comment your definition of complex can also be used!

Clear[complex];
complex = PL /. sol;
Plot[{complex, Hold[Evaluate@D[complex, r]]} // Release, {r, 0.1, 2}]

In the mean time don't forget to notice the Hold issue with the Plot command. To perform symbolic operations (e.g Differentiation) inside Plot you need to use Hold as Plot replaces the function variable with numeric arguments and thus can result into wrong result for the inner symbolic operation.

Answer 2

Using your definitions, you can simply write:

Plot[Evaluate@{complex, D[complex, r]}, {r, 0.1, 2}]

The Evaluate makes sure that D[complex, r] gets evaluated and the expression for the derivative is passed to Plot.