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Strictly speaking, the $x$ direction derivative should be the difference between the left and right pixel of each pixel. So, then I should be using a $1\times 3$ filter:

\begin{bmatrix}-1&0&1\end{bmatrix}

If that is so, why is the Sobel operator a $3\times 3$? \begin{bmatrix}-1&0&1\\-2&0&2\\-1&0&1\end{bmatrix}

Is it so that the approximation of the derivative is less susceptible to noise?

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Indeed, it adds smoothing in the $y$ direction. The Sobel filter is the separable combination of the centered derivative $[−1,\;0,\;1]$ along $x$, and the $3$-point binomial smoother $[1,\;2,\;1]$ along $y$.

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