Question

Determine the 99th derivative of y=cosx

Answer 1

Do you understand the way the derivatives will cycle when we deal with cosine?
Every 4th derivative will bring us back to cosine.

So if we want the 99th derivative, we want the (100-1)th derivative.
What is the 100th derivative?
Well, 100 is a multiple of 4. (25 times 4 ).

So 100 derivatives would be that process of taking the derivative 4 times, and getting cosine back again ( repeating that process 25 times).
So we would end up back at cosine.

So we want the derivative just before that.

(cosx)' = -sinx
(-sinx)' = -cosx
(-cosx)' = sinx
(sinx)' = cosx <-- this is what we get when we apply 4 derivatives, or any multiple of 4 derivatives. What did the derivative produce just before this 4th derivative?