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(A) The greatest integer function $[x]$ is a step function that takes integer values for all $x \in \mathbb{R}$.For any integer $n$,$[x] = n$ in the i

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$0$

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(A) The greatest integer function $[x]$ is a step function that takes integer values for all $x \in \mathbb{R}$.For any integer $n$,$[x] = n$ in the interval $[n, n+1)$.At $x = -1$,we consider the left-hand and right-hand derivatives.For $x$ in the neighborhood of $-1$,specifically for $x \in [-1, 0)$,$[x] = -1$.Thus,for $x \in [-1, 0)$,the function $f(x) = \sin(\pi[x]) = \sin(\pi(-1)) = \sin(-\pi) = 0$.Since the function is constant $(0)$ in the interval $[-1, 0)$,its derivative $\frac{d}{dx} \sin(\pi[x])$ is $0$ at $x = -1$ (considering the right-hand derivative).Since the function is constant in the interval $[-2, -1)$,$[x] = -2$,so $f(x) = \sin(-2\pi) = 0$.Therefore,the derivative is $0$.

$[x]$ represents the greatest integer function. At $x = -1$,what is the value of $\frac{d}{dx} \sin(\pi[x])$?

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