Sobel Vs Gaussian Derivative
The Sobel kernels can also be thought of as $3 × 3$ approximations to first-derivative-of-Gaussian kernels.

The Sobel filter is nothing but jointly applying the following:


Smoothing in the perpendicular direction of the derivative such as $h_s=[1,2,1]^T$ 
Simple central difference in the derivative direction such as $h_d=[1,0,-1]$


Then $$h_{sobel} = h_s  h_d$$

The smoothing factor is an approximate triangle shaped filter. A Gaussian is naturally a better replacement. In fact, if larger sizes of Sobel is desired, people first smooth the image with a Gaussian filter, then apply the Sobel kernel as is. That's because:

$$
\frac{\partial}{\partial x} {(I*G)} = I*\frac{\partial}{\partial x} {(G)}
$$

This approximation enables the Sobel operator to be computed quite quickly, while Gaussian derivatives might be more accurate.