If we have $f(x,y)=f(\overline{x}\pm\sigma_x,\overline{y}\pm\sigma_y)$ how is then $$\sigma_f=\sqrt{\left(\frac{\partial{f}}{\partial{x}}\right)^2\sigma_x^2+\left(\frac{\partial{f}}{\partial{y}}\right)^2\sigma_y^2}~~~?$$
I have looked at this Wikipedia page but I don't quite understand this approach since I'm not familiar with approximating functions with Taylor series. I have also found that in this video it is said that that way of calculating the standard deviation for multivariate functions somehow gives us a average standard deviation but I don't understand why/how.
