Expert: Ahmed Salami - 6/6/2009

Question
If f(x) = sinx+cosx, find the value of n such that f^n(x) = f(x).  I'm not sure
really what its asking or how to get it started.  Thanks.

Answer
Hi Patrick,
What the question is asking is how many times would you differentiate f(x) to arrive at a function exactly equal to f(x). f^n(x) is simply the nth derivative of f(x).
To find out, you keep on differentiating until you arrive back at sinx + cosx. I would advise you to do that and compare with the more advanced solution below.
If g(x) = sinx
g^n(x) = sin(x + nπ/2)
and it always returns to sinx whenever n/2 is even, or simply when n is a multiple of 4. Its a similar case for cosx too and so the values of n are all multiples of 4.

Regards