So I noticed I have some wrong intuition about partial derivatives and I would appreciate someone correcting me why I am wrong. Say we have a function $f(x,y)=2\sqrt{xy}$. Clearly $f_x(x,y)$ decreases when x increases and $f_y(x,y)$ decreases when y increases. Now my intuition was telling me that this has to mean that every partial derivative in the direction that increases $x$ and $y$ should also decrease. However, if we consider $f(x,x)=2x$ this function actually increases at a constant rate when both $x$ and $y=x$ increase. I thought this since we know that partial in any direction is $<f_x,f_y>\cdot u$ where u is the unit vector of the direction. But then a linear combination of decreasing functions, is decreasing, right? This seems to contradict the fact that $f(x,x)=2x$
So where is my blunder? Why was I wrong?
thank you