# Math Help - What Is The nth Derivative of sin(x)

1. ## What Is The nth Derivative of sin(x)

How can you write out the nth derivative of sin(x)?

2. Let $f(x)=\sin x$

$f'(x)=\cos x$

$f''(x)=-\sin x$

$f'''(x)=-\cos x$

$f''''(x)=\sin x$ and then the cycle repeats

So you could just write it out using if's and keep dividing the n of $f^{(n)}$by 4 to get the remainder

But upon further analysis, we realize

$f'(x)=\cos x=\sin (\frac{\pi}{2}+x)$

$f''(x)=-\sin x=\sin(\frac{2\pi}{2}+x)$

$f'''(x)=-\cos x=\sin(\frac{3\pi}{2}+x)$

$f''''(x)=\sin x=\sin(\frac{4\pi}{2}+x)$

So $f^{(n)}(x)=\sin(\frac{n\pi}{2}+x)$

3. Works for me, thanks!