Exercise. Find the first nine derivatives of the function f (x) = sin x.
The derivatives of f (x) = sin x follow a pattern. Every 4 derivatives, we find the original function. f (x) is the same as f (4)(x) is the same as f 8(x).
Similarly, f (1)(x) is the same as f (5)(x) is the same as f 9(x) .
If two numbers differ by a multiple of 4, the derivatives of sin x corresponding to those numbers will be the same.