YES

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))))

linear polynomial interpretations on N:

  dx_A(x1) = 1
  dx#_A(x1) = 4
  one_A = 1
  one#_A = 0
  a_A = 1
  a#_A = 6
  zero_A = 0
  zero#_A = 5
  plus_A(x1,x2) = 1
  plus#_A(x1,x2) = 3
  times_A(x1,x2) = 1
  times#_A(x1,x2) = 3
  minus_A(x1,x2) = 1
  minus#_A(x1,x2) = 1
  neg_A(x1) = 1
  neg#_A(x1) = 0
  div_A(x1,x2) = 1
  div#_A(x1,x2) = 2
  exp_A(x1,x2) = x1 + 1
  exp#_A(x1,x2) = 1
  two_A = 1
  two#_A = 1
  ln_A(x1) = x1 + 1
  ln#_A(x1) = 0

precedence:

  times > dx = a > zero = plus = minus = neg = exp > one = ln > div > two