YES

TRS:

  din(der(plus(X,Y))) -> u21(din(der(X)),X,Y)
  u21(dout(DX),X,Y) -> u22(din(der(Y)),X,Y,DX)
  u22(dout(DY),X,Y,DX) -> dout(plus(DX,DY))
  din(der(times(X,Y))) -> u31(din(der(X)),X,Y)
  u31(dout(DX),X,Y) -> u32(din(der(Y)),X,Y,DX)
  u32(dout(DY),X,Y,DX) -> dout(plus(times(X,DY),times(Y,DX)))
  din(der(der(X))) -> u41(din(der(X)),X)
  u41(dout(DX),X) -> u42(din(der(DX)),X,DX)
  u42(dout(DDX),X,DX) -> dout(DDX)

linear polynomial interpretations on N:

  din_A(x1) = 1
  din#_A(x1) = 1
  der_A(x1) = x1 + 1
  der#_A(x1) = 3
  plus_A(x1,x2) = x1 + 1
  plus#_A(x1,x2) = 5
  u21_A(x1,x2,x3) = x1
  u21#_A(x1,x2,x3) = 4
  dout_A(x1) = 4
  dout#_A(x1) = 0
  u22_A(x1,x2,x3,x4) = x1 + 3
  u22#_A(x1,x2,x3,x4) = x1 + 2
  times_A(x1,x2) = x1 + x2 + 1
  times#_A(x1,x2) = 5
  u31_A(x1,x2,x3) = x1
  u31#_A(x1,x2,x3) = 4
  u32_A(x1,x2,x3,x4) = 4
  u32#_A(x1,x2,x3,x4) = x1 + 2
  u41_A(x1,x2) = x1
  u41#_A(x1,x2) = x1 + 1
  u42_A(x1,x2,x3) = 4
  u42#_A(x1,x2,x3) = 4

precedence:

  times > der > u31 > u32 > plus > u21 > din = u22 > u41 > u42 > dout