Question 0. Let's put the issue to <I>Mathematica</I>:

Question 0. Let's put the issue to Mathematica:

D[x Cos[xy],x]

D[xCos[x*y],x]

D[x Cos(x y),x]

D[x Cos[x y],x]

Cos[xy]
;[o]
Cos[xy]

General::spell1:
Possible spelling error: new symbol name "xCos"
is similar to existing symbol "Cos".

y*Derivative[1][xCos][x*y]
;[o]
y xCos'[x y]

Syntax::bktwrn:
Warning: "Cos(x y)" should probably be "Cos[x y]".

2*Cos*x*y
;[o]
2 Cos x y

Cos[x*y] - x*y*Sin[x*y]
;[o]
Cos[x y] - x y Sin[x y]

Only the last one of these is right.