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))))

max/plus interpretations on N:

  dx_A(x1) = max{34, 11 + x1}
  dx#_A(x1) = max{34, 11 + x1}
  one_A = 14
  one#_A = 14
  a_A = 1
  a#_A = 1
  zero_A = 0
  zero#_A = 0
  plus_A(x1,x2) = max{34, x1, x2}
  plus#_A(x1,x2) = max{34, x1, x2}
  times_A(x1,x2) = max{17, 7 + x1, x2}
  times#_A(x1,x2) = max{17, 7 + x1, x2}
  minus_A(x1,x2) = max{14, x1, x2}
  minus#_A(x1,x2) = max{14, x1, x2}
  neg_A(x1) = max{22, x1}
  neg#_A(x1) = max{22, x1}
  div_A(x1,x2) = max{12, x1, 19 + x2}
  div#_A(x1,x2) = max{12, x1, 19 + x2}
  exp_A(x1,x2) = max{15, 10 + x1, 13 + x2}
  exp#_A(x1,x2) = max{15, 10 + x1, 13 + x2}
  two_A = 2
  two#_A = 2
  ln_A(x1) = max{20, 9 + x1}
  ln#_A(x1) = max{20, 9 + x1}

precedence:

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