Question

$\frac{d^{91}}{dx^{91}}(\sin(x))$
$-\sin(x)$
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Explanation:

Step1: Recall derivative pattern of $\sin(x)$

The first - derivative of $y = \sin(x)$ is $y'=\cos(x)$, the second - derivative $y''=-\sin(x)$, the third - derivative $y''' = -\cos(x)$, and the fourth - derivative $y^{(4)}=\sin(x)$. The pattern repeats every 4 derivatives.

Step2: Divide the order of derivative by 4

We want to find the 91 - st derivative of $\sin(x)$. Divide 91 by 4: $91\div4 = 22\cdots\cdots3$.

Step3: Determine the 91 - st derivative

Since the remainder is 3 when 91 is divided by 4, the 91 - st derivative of $\sin(x)$ has the same form as the third - derivative of $\sin(x)$. So, $\frac{d^{91}}{dx^{91}}(\sin(x))=-\cos(x)$.

Answer:

$-\cos(x)$