TITLE: Why does the generalised derivative have to be a linear transformation?
QUESTION [24 upvotes]: I am starting to learn Real Analysis and I have come across the generalised definition of the derivative for higher dimensions. I realise that the derivative being a linear transformation nicely accommodates the one dimensional case where the derivative is just a constant at any point. I also understand it can't be as simple as multiplication by a constant for higher dimensions since you can approach a point along multiple curves in higher dimensions. But where did we hit upon the fact that it has to be linear? Why couldn't it be some other type of function? I would like to get an intuitive explanation.

REPLY [5 votes]: I'm going to give a very intuitive non-rigorous reason.

The derivative captures the meaning of relative rates of change of variables near some point. What this effectively means is that when you zoom in to that point, the constraint surface gets closer and closer to being a (hyper-)plane. This is especially so when we want the rate of change of some dependent variable with respect to multiple independent variables, since we expect it to change smoothly with respect to each independent variable. Zoom in sufficiently, and all the individual smoothness requirements will imply that the dependent variable varies linearly with respect to the independent variables near the point under consideration.