I am trying to find the 2-D derivative of an elongated Gaussian density. The Gaussian has standard deviations $\sigma_x$ and $\sigma_y$. How can I get the scale-normalized 2-D Gaussian derivative in this case? Normally, I'd multiply by $\sigma^2$, but what to do when the standard deviation in different in each dimension? Thank you.
If I understand the question, you have three options:
- You can take the partial derivative with respect to either variable.
- You can take the total derivative with respect to another variable on which they both depend.
- You can take the gradient, which is a vector whose components are the partial derivatives of the components.
The problem should determine the appropriate choice.
If you differentiate in the $x$ direction, due to the separability, you simply replace the scale factor $\sigma$ that goes out of the exponential by $\sigma_x$ (potentially with its square), the same in the $y$-direction. If you differentiate two times, you will get some $\sigma_x^2\sigma_y^2$, $\sigma_x^4$ or $\sigma_y^4$ instead of $\sigma^4$.
Your formula should be consistent with the former ones when you set $\sigma_x = \sigma_y = \sigma$.