Dt¶

Total derivative treating all variables as potentially dependent.

$ wo 'Dt[5, x]'
0

Dt[expr, {x, n}] differentiates n times:

$ wo 'Dt[x^4, {x, 3}]'
24*x

$ wo 'Dt[Sin[x], {x, 2}]'
-Sin[x]

A symbolic count cannot be carried out, so the call is held — this is how Rodrigues's formulas are written:

$ wo 'Dt[(x^2 - 1)^n, {x, n}]'
Dt[(-1 + x^2)^n, {x, n}]

Other variables stay dependent, and their own derivatives grow an argument rather than nesting — repeats folding into the {x, n} order spec:

$ wo 'Dt[x y, {x, 2}]'
2*Dt[y, x] + x*Dt[y, {x, 2}]

A held total derivative does not depend on the symbol it differentiates, so differentiating it by that symbol again gives 0:

$ wo 'Dt[Dt[y, x], y]'
0

Dt[expr, x1, x2, ...] differentiates against each variable in turn:

$ wo 'Dt[x^3, x, x]'
6*x