Termination w.r.t. Q of the given QTRS could be proven:

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(minus(x)) → minus(D(x))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Combined order from the following AFS and order.
D(x1)  =  D(x1)
t  =  t
1  =  1
constant  =  constant
0  =  0
+(x1, x2)  =  +(x1, x2)
*(x1, x2)  =  *(x1, x2)
-(x1, x2)  =  -(x1, x2)
minus(x1)  =  x1
div(x1, x2)  =  div(x1, x2)
pow(x1, x2)  =  pow(x1, x2)
2  =  2
ln(x1)  =  ln(x1)

Recursive path order with status [RPO].
Quasi-Precedence:

D₁ > 1 > pow₂
D₁ > 0 > pow₂
D₁ > +₂ > pow₂
D₁ > -₂ > pow₂
D₁ > [div₂, 2] > *₂ > pow₂
D₁ > ln₁ > pow₂
t > 1 > pow₂
constant > pow₂

Status:

ln₁: multiset
t: multiset
-₂: multiset
D₁: multiset
2: multiset
pow₂: multiset
0: multiset
+₂: [2,1]
constant: multiset
div₂: multiset
*₂: multiset
1: multiset

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

D(t) → 1
D(constant) → 0
D(+(x, y)) → +(D(x), D(y))
D(*(x, y)) → +(*(y, D(x)), *(x, D(y)))
D(-(x, y)) → -(D(x), D(y))
D(div(x, y)) → -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
D(ln(x)) → div(D(x), x)
D(pow(x, y)) → +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

D(minus(x)) → minus(D(x))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(D(x₁)) = 2 + 2·x₁   
POL(minus(x₁)) = 1 + x₁   

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

D(minus(x)) → minus(D(x))

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE

(7) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(8) TRUE

(9) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(10) TRUE