One of the first trig formulas you see in a Calculus book is " ". There are a number of ways of showing that, depending upon exactly how you define "sin(x)".
With , the difference quotient is . Using the fact that , we can write that as . So the derivative is . Notice that we were able to take x completely out of the limit so the derivative is the original function, , times some constant. How you take that limit to find the constant depends upon exactly how you want to handle exponentials but it comes out to be the natural logarithm of .85.