YES

Problem 1: 

(VAR f g)
(THEORY 
(AC * +))
(RULES
dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
dx(+(f,g)) -> +(dx(f),dx(g))
dx(-(f,g)) -> -(dx(f),dx(g))
dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
dx(0) -> 0
dx(1) -> 0
dx(X) -> 1
dx(a) -> 0
dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
dx(ln(f)) -> /(dx(f),f)
dx(neg(f)) -> neg(dx(f))
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 *#(*(x2,x3),x4) = *#(x2,*(x3,x4))
 *#(x2,x3) = *#(x3,x2)
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 DX(*(f,g)) -> DX(f)
 DX(*(f,g)) -> DX(g)
 DX(+(f,g)) -> DX(f)
 DX(+(f,g)) -> DX(g)
 DX(-(f,g)) -> DX(f)
 DX(-(f,g)) -> DX(g)
 DX(/(f,g)) -> DX(f)
 DX(/(f,g)) -> DX(g)
 DX(exp(f,g)) -> DX(f)
 DX(exp(f,g)) -> DX(g)
 DX(ln(f)) -> DX(f)
 DX(neg(f)) -> DX(f)
-> EAxioms:
 *(*(x2,x3),x4) = *(x2,*(x3,x4))
 *(x2,x3) = *(x3,x2)
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
 dx(+(f,g)) -> +(dx(f),dx(g))
 dx(-(f,g)) -> -(dx(f),dx(g))
 dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
 dx(0) -> 0
 dx(1) -> 0
 dx(X) -> 1
 dx(a) -> 0
 dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
 dx(ln(f)) -> /(dx(f),f)
 dx(neg(f)) -> neg(dx(f))
-> SRules:
 *#(*(x2,x3),x4) -> *#(x2,x3)
 *#(x2,*(x3,x4)) -> *#(x3,x4)
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)

Problem 1: 

SCC Processor:
-> FAxioms:
 *#(*(x2,x3),x4) = *#(x2,*(x3,x4))
 *#(x2,x3) = *#(x3,x2)
 +#(+(x2,x3),x4) = +#(x2,+(x3,x4))
 +#(x2,x3) = +#(x3,x2)
-> Pairs:
 DX(*(f,g)) -> DX(f)
 DX(*(f,g)) -> DX(g)
 DX(+(f,g)) -> DX(f)
 DX(+(f,g)) -> DX(g)
 DX(-(f,g)) -> DX(f)
 DX(-(f,g)) -> DX(g)
 DX(/(f,g)) -> DX(f)
 DX(/(f,g)) -> DX(g)
 DX(exp(f,g)) -> DX(f)
 DX(exp(f,g)) -> DX(g)
 DX(ln(f)) -> DX(f)
 DX(neg(f)) -> DX(f)
-> EAxioms:
 *(*(x2,x3),x4) = *(x2,*(x3,x4))
 *(x2,x3) = *(x3,x2)
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
 dx(+(f,g)) -> +(dx(f),dx(g))
 dx(-(f,g)) -> -(dx(f),dx(g))
 dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
 dx(0) -> 0
 dx(1) -> 0
 dx(X) -> 1
 dx(a) -> 0
 dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
 dx(ln(f)) -> /(dx(f),f)
 dx(neg(f)) -> neg(dx(f))
-> SRules:
 *#(*(x2,x3),x4) -> *#(x2,x3)
 *#(x2,*(x3,x4)) -> *#(x3,x4)
 +#(+(x2,x3),x4) -> +#(x2,x3)
 +#(x2,+(x3,x4)) -> +#(x3,x4)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 DX(*(f,g)) -> DX(f)
 DX(*(f,g)) -> DX(g)
 DX(+(f,g)) -> DX(f)
 DX(+(f,g)) -> DX(g)
 DX(-(f,g)) -> DX(f)
 DX(-(f,g)) -> DX(g)
 DX(/(f,g)) -> DX(f)
 DX(/(f,g)) -> DX(g)
 DX(exp(f,g)) -> DX(f)
 DX(exp(f,g)) -> DX(g)
 DX(ln(f)) -> DX(f)
 DX(neg(f)) -> DX(f)
-> FAxioms:
 *(*(x2,x3),x4) -> *(x2,*(x3,x4))
 *(x2,x3) -> *(x3,x2)
 +(+(x2,x3),x4) -> +(x2,+(x3,x4))
 +(x2,x3) -> +(x3,x2)
-> EAxioms:
 *(*(x2,x3),x4) = *(x2,*(x3,x4))
 *(x2,x3) = *(x3,x2)
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
->->-> Rules:
 dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
 dx(+(f,g)) -> +(dx(f),dx(g))
 dx(-(f,g)) -> -(dx(f),dx(g))
 dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
 dx(0) -> 0
 dx(1) -> 0
 dx(X) -> 1
 dx(a) -> 0
 dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
 dx(ln(f)) -> /(dx(f),f)
 dx(neg(f)) -> neg(dx(f))
-> SRules:
 Empty

Problem 1: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 DX(*(f,g)) -> DX(f)
 DX(*(f,g)) -> DX(g)
 DX(+(f,g)) -> DX(f)
 DX(+(f,g)) -> DX(g)
 DX(-(f,g)) -> DX(f)
 DX(-(f,g)) -> DX(g)
 DX(/(f,g)) -> DX(f)
 DX(/(f,g)) -> DX(g)
 DX(exp(f,g)) -> DX(f)
 DX(exp(f,g)) -> DX(g)
 DX(ln(f)) -> DX(f)
 DX(neg(f)) -> DX(f)
-> EAxioms:
 *(*(x2,x3),x4) = *(x2,*(x3,x4))
 *(x2,x3) = *(x3,x2)
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
 dx(+(f,g)) -> +(dx(f),dx(g))
 dx(-(f,g)) -> -(dx(f),dx(g))
 dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
 dx(0) -> 0
 dx(1) -> 0
 dx(X) -> 1
 dx(a) -> 0
 dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
 dx(ln(f)) -> /(dx(f),f)
 dx(neg(f)) -> neg(dx(f))
-> SRules:
 Empty
->Projection:
 pi(DX) = [1]

Problem 1: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 *(*(x2,x3),x4) = *(x2,*(x3,x4))
 *(x2,x3) = *(x3,x2)
 +(+(x2,x3),x4) = +(x2,+(x3,x4))
 +(x2,x3) = +(x3,x2)
-> Rules:
 dx(*(f,g)) -> +(*(dx(f),g),*(dx(g),f))
 dx(+(f,g)) -> +(dx(f),dx(g))
 dx(-(f,g)) -> -(dx(f),dx(g))
 dx(/(f,g)) -> -(/(dx(f),g),/(*(dx(g),f),exp(g,2)))
 dx(0) -> 0
 dx(1) -> 0
 dx(X) -> 1
 dx(a) -> 0
 dx(exp(f,g)) -> +(*(dx(f),*(exp(f,-(g,1)),g)),*(dx(g),*(exp(f,g),ln(f))))
 dx(ln(f)) -> /(dx(f),f)
 dx(neg(f)) -> neg(dx(f))
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.