<s>[INST] $99$th derivative of $\sin x$ Can someone help me calculate the $99$th derivative of $\sin(x)$?
Calculate $f^{(99)}(x) $ for the function $f(x) = \sin(x) $
 [/INST] Notice if $f(x) = \sin x$
$f'(x) = \cos x = \sin( x + \frac{\pi}{2}) $
$f''(x) = - \sin x = \sin( x + \pi) = \sin( x + 2 (\frac{\pi}{2})) $
$f'''(x) = - \cos x = \sin(x + 3( \frac{\pi}{2} ))$
$f''''(x) = \sin x $
Hence, we can say that 
$$ f^{(n)} (x) = \sin \left( x + n \cdot\frac{\pi}{2} \right) $$
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