Question

Find f^(xxvi)(x) if f (x) = sin x,
So its asking for the 26th derivative.

Answer 1

The first five derivatives is the following list:
{cos(x),-sin(x),-cos(x),sin(x),cos(x)}
Mod(26,4)=2
The 26th derivative is -sin[x]

Answer 2

\(f(x)=\sin{x}\) \(f'(x)=\cos{x}\)
\(f''(x)=-\sin{x}\) \(f^{(3)}=-\cos{x}\)
\(f^{(4)}=\sin{x}\)
From this pattern you can see that the derivatives repeat themselves after each four derivatives, Thus \(f^{(26)}=f^{(2)}=-\sin{x}\).

Answer 3

Thanks. I've been getting these wrong for the stupid mistake that I started the sequence at sin x insted of cosine