Derivatives of Trigonometric functions

Question

10. Find the higher derivative d^2017/ dx^2017  (2 cos x) by finding the first eight derivatives and ob-

serving the pattern that occurs.

Answer 1

Emery,

Hopefully you attempted to find the first eight derivatives. If so, you would have got:

f(x) = 2 cos x

f'(x) = -2sin x

f''(x) = -2cos x

f'''(x) = 2sinx x

f(''''x) = 2 cos x

f'''''(x) = -2sin x

f''''''(x) = -2cos x

f'''''''(x) = 2sinx x

f''''''''(x) = 2cos x

At this point, hopefully you can see a pattern and only 4 possible answers not matter how many times we take the derivative. It might help to rearrange them this way, with the answers as our columns and the nth derivative numbers under them:

2cos x -2sin x -2cos x 2sin x

0 1 2 3

4 5 6 7

8...

This pattern will continue indefinitely. So we don't need to complete 2017 derivatives to find our answer. All you need to do is divide the nth derivative by 4 and see whether your answer has a remainder. No remainder, your answer is in the "0-column" (2cos x), a remainder of 1, the "1-column" (-2sin x), a remainder of 2, the "2-column" (-2cos x), and a remainder of 3, in the last "3-column" (2sin x). So in our case, dividing 2017/4 gives us 504 with a remainder of 1. So our derivative ends up in the "1-column", -2sin x.

I hope that helps.