What is $n$-th derivative of $\cos(x)$?

Question

There is a question on this website which tries to prove that the value of nth derivative of $\cos(x)$ is 'something'.

The question is given below:

What if I were to find the value of $n$-th derivative rather than prove it using induction and other methods.

How to 'FIND THE VALUE OF IT'?

See https://math.stackexchange.com/questions/1752455/100-th-derivative-of-the-function-fx-ex-cosx/1752462#1752462 — lab bhattacharjee, Jul 04 '18 at 07:39
how would you want to find the value of $n$ as $n$ is not defined? — Noa Even, Jul 04 '18 at 09:03

score 6 · Answer 1 · answered Jul 04 '18 at 07:38

Try the first $n^{th}$ derivatives and observe the pattern. You get

$$\cos x,-\sin x,-\cos x,\sin x$$ and so on periodically.

If you observe this on the trigonometric circle, you will notice rotations by $\frac\pi2$ radians each time, so that you can summarize as

$$\cos\left(x+n\frac\pi2\right).$$

A more "advanced" way is by means of complex numbers. We have

$$\cos x=\Re e^{ix},$$

then

$$(e^{ix})^{(n)}=i^ne^{ix}=e^{i(x+n\pi/2)}.$$

1 Answers1