I know how to prove this via limits and I'm okay with that. What I want to understand is the interpretation of the theorem and specifically a visualisation of why what the theorem states must be the case. My guess is that this theorem is saying that change is symmetrical. But I don't know if this is only true for second derivatives. If you don't know this theorem by its name the theorem basically says this: ∂/∂y(∂f/∂x)=∂/∂x(∂f/∂y) Also, I would like to know if you consider my focus on visualisation to be not worthwhile and that I should instead just trust this theorem. I thank you in advance.