Write a Sine-Deriving Machine

Python, 42 bytes

from math import*
lambda x,n:sin(x+pi/2*n)

Uses the fact that differentiating shifts the function by pi/2 left.

Actually, 5 bytes

╦*½+S

Try it online

Takes input as t\nx.

Explanation:

╦*½+S
╦*     pi*n
  ½    half that
   +   add t
    S  sine

C (gcc), 30 bytes

#define f(x,n)sin(x+n*acos(0))