Binomial theorem

 binomial theorem.....and sin and cos  substitution for y and x in ( x + y)^n expansion ....then because the  function's is assumed a continuous differentiable for reducing and increasing powers expressed in the derivative and antiderivative representations......... captured to ...... n(x + y )^n-1 + 0 and  (x + y)^n + z calculus transformation. The action of continuous differentiation terminates when the nth derivative of  ( x + y)^n = 0 and (n-1)^th derivative of  ( x + y)^n = const = Pn , given Pn is the n^th successor prime number P.