Why is "1" not a valid variable? Why is there not a function plotted in the Plot function? [duplicate]

Question

This is my code for playing this kind of stuff

    A = {1, 1};
    B = {0, -1};
    c = {x, 0};
    l1 = EuclideanDistance[A, c];
    l2 = EuclideanDistance[c, B];
    f[x_] := l1/1 + l2/2;
    D[f[x], x]
    g[x_] := D[f[x], x]
    g[1]
    Plot[g[x], {x, -20, 20}, PlotRange -> 20 {{-1, 1}, {-1, 1}}, 
    PlotStyle -> Red, AspectRatio -> Automatic, ImageSize -> 400]

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To clarify, there seem to be three or four questions, some of which are in the title, some to be inferred from comments or evaluating the code:

1. Why are Abs'[1 - x] and Abs'[x] left unevaluated in the output of D[f[x], x]?
2. Why does g[1] result in an error "General::ivar 1 is not a valid variable"?
3. Why does Plot[...] generate similar General::ivar errors?
4. Why does the Plot comes up empty when only the previous two problems are fixed? (It came up blank originally because of the problem with D[...].)

Answer 1

There are several things going on here. First, both l1 and l2 use Abs:

l1
l2

Sqrt[1 + Abs[1 - x]^2]

Sqrt[1 + Abs[x]^2]

Mathematica will not compute the derivative of Abs because it is not analytic as a complex function. One possibility is to use ComplexExpand to eliminate the Abs:

m1 = ComplexExpand @ l1
m2 = ComplexExpand @ l2

Sqrt[1 + (1 - x)^2]

Sqrt[1 + x^2]

Next, your definition of f is problematic because the RHS does not contain an explicit x variable. It is better to use either:

f[x_] = m1/2 + m2/2

or:

f[x_] := Evaluate[m1/2 + m2/2]

so that the RHS is evaluated when defining f. Finally, when you define:

g[x_] := D[f[x], x]

and evaluate g[1], Mathematica will try to evaluate:

D[f[1], 1]

which is why you get your message. It is much better to use:

g = f'

instead to define a derivative function. So:

Clear[f, g];

f[x_] = m1/2 + m2/2;
g = f';
Plot[g[x], {x, -20, 20}]