YES

Problem:
 dx(X) -> one()
 dx(a()) -> zero()
 dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
 dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
 dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
 dx(neg(ALPHA)) -> neg(dx(ALPHA))
 dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
 dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
 dx(exp(ALPHA,BETA)) ->
 plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times
                                                                (exp(ALPHA,BETA),
                                                                 times
                                                                 (ln(ALPHA),dx(BETA))))

Proof:
 DP Processor:
  DPs:
   dx#(plus(ALPHA,BETA)) -> dx#(BETA)
   dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
   dx#(times(ALPHA,BETA)) -> dx#(BETA)
   dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
   dx#(minus(ALPHA,BETA)) -> dx#(BETA)
   dx#(minus(ALPHA,BETA)) -> dx#(ALPHA)
   dx#(neg(ALPHA)) -> dx#(ALPHA)
   dx#(div(ALPHA,BETA)) -> dx#(BETA)
   dx#(div(ALPHA,BETA)) -> dx#(ALPHA)
   dx#(ln(ALPHA)) -> dx#(ALPHA)
   dx#(exp(ALPHA,BETA)) -> dx#(BETA)
   dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
  TRS:
   dx(X) -> one()
   dx(a()) -> zero()
   dx(plus(ALPHA,BETA)) -> plus(dx(ALPHA),dx(BETA))
   dx(times(ALPHA,BETA)) -> plus(times(BETA,dx(ALPHA)),times(ALPHA,dx(BETA)))
   dx(minus(ALPHA,BETA)) -> minus(dx(ALPHA),dx(BETA))
   dx(neg(ALPHA)) -> neg(dx(ALPHA))
   dx(div(ALPHA,BETA)) -> minus(div(dx(ALPHA),BETA),times(ALPHA,div(dx(BETA),exp(BETA,two()))))
   dx(ln(ALPHA)) -> div(dx(ALPHA),ALPHA)
   dx(exp(ALPHA,BETA)) ->
   plus(times(BETA,times(exp(ALPHA,minus(BETA,one())),dx(ALPHA))),times
                                                                  (exp(ALPHA,BETA),
                                                                   times
                                                                   (ln(ALPHA),dx(BETA))))
  Usable Rule Processor:
   DPs:
    dx#(plus(ALPHA,BETA)) -> dx#(BETA)
    dx#(plus(ALPHA,BETA)) -> dx#(ALPHA)
    dx#(times(ALPHA,BETA)) -> dx#(BETA)
    dx#(times(ALPHA,BETA)) -> dx#(ALPHA)
    dx#(minus(ALPHA,BETA)) -> dx#(BETA)
    dx#(minus(ALPHA,BETA)) -> dx#(ALPHA)
    dx#(neg(ALPHA)) -> dx#(ALPHA)
    dx#(div(ALPHA,BETA)) -> dx#(BETA)
    dx#(div(ALPHA,BETA)) -> dx#(ALPHA)
    dx#(ln(ALPHA)) -> dx#(ALPHA)
    dx#(exp(ALPHA,BETA)) -> dx#(BETA)
    dx#(exp(ALPHA,BETA)) -> dx#(ALPHA)
   TRS:
    
   Arctic Interpretation Processor:
    dimension: 1
    usable rules:
     
    interpretation:
     [dx#](x0) = x0,
     
     [ln](x0) = 1x0,
     
     [exp](x0, x1) = 1x0 + 2x1 + 3,
     
     [div](x0, x1) = 1x0 + 1x1 + 3,
     
     [neg](x0) = 4x0 + 0,
     
     [minus](x0, x1) = 1x0 + 4x1 + 0,
     
     [times](x0, x1) = 1x0 + 1x1 + 6,
     
     [plus](x0, x1) = 1x0 + 2x1 + 1
    orientation:
     dx#(plus(ALPHA,BETA)) = 1ALPHA + 2BETA + 1 >= BETA = dx#(BETA)
     
     dx#(plus(ALPHA,BETA)) = 1ALPHA + 2BETA + 1 >= ALPHA = dx#(ALPHA)
     
     dx#(times(ALPHA,BETA)) = 1ALPHA + 1BETA + 6 >= BETA = dx#(BETA)
     
     dx#(times(ALPHA,BETA)) = 1ALPHA + 1BETA + 6 >= ALPHA = dx#(ALPHA)
     
     dx#(minus(ALPHA,BETA)) = 1ALPHA + 4BETA + 0 >= BETA = dx#(BETA)
     
     dx#(minus(ALPHA,BETA)) = 1ALPHA + 4BETA + 0 >= ALPHA = dx#(ALPHA)
     
     dx#(neg(ALPHA)) = 4ALPHA + 0 >= ALPHA = dx#(ALPHA)
     
     dx#(div(ALPHA,BETA)) = 1ALPHA + 1BETA + 3 >= BETA = dx#(BETA)
     
     dx#(div(ALPHA,BETA)) = 1ALPHA + 1BETA + 3 >= ALPHA = dx#(ALPHA)
     
     dx#(ln(ALPHA)) = 1ALPHA >= ALPHA = dx#(ALPHA)
     
     dx#(exp(ALPHA,BETA)) = 1ALPHA + 2BETA + 3 >= BETA = dx#(BETA)
     
     dx#(exp(ALPHA,BETA)) = 1ALPHA + 2BETA + 3 >= ALPHA = dx#(ALPHA)
    problem:
     DPs:
      
     TRS:
      
    Qed