Vtyjkyuk

Say we are looking for $fracdelta xdelta vecy$ where $vecy$ is a vector. $fracdelta xdelta vecy$ is the derivative of $x$ wrt a vector $vecy$. If $x in mathbbR$ then I suppose we can say where $fracdelta xdelta vecy = 0$ we have the point in that space where x is maximised/minimised. However, what happens if x is a vector? What is the intuitive meaning then?

If I understand your question correctly, when you say $fracdxdy$, $x$ is some function (whose value changes according to where you look at it on its domain), yes? Therefore, the derivative of $x$ with respect to $y$ tells you the rate-of-change of $x$ in the direction of $y$ ($y$, as you said, is a vector). That is, if you were to look at the domain of $x$ along the vector $y$, you get the rate of change of $x$ with respect to $y$. (So when that derivative reaches 0, we have a maximum or a minimum rate-of-change of $x$ in the direction of $y$.)

Now, note I'm using the term "rate-of-change" pretty often. If $x$ is a function, whose value changes according to where you are on its domain, then $x$ clearly has some rate of change.

But, if $x$ is a vector, then... is it changing at all? (Think about it before you read further!)

Nope! The derivative of a vector is always $0$—because vectors don't change. It's just a static object. So the derivative of a vector is $0$ with respect to anything, including other vectors.