Visualizing the total differential I'm trying to convince myself of the fact that $d z=\frac{\partial f}{\partial x} d x+\frac{\partial f}{\partial y} d y$ by looking at this picture (i.e. $dz$ should equal the sum of the two segments in red), but can't seem to do it. I guess I can go the difficult road, trying to compute $dz$ by using the Pythagorean formula on the diagonal of the square on top of the cube and the diagonal of the yellow parallelogram. But is there an easier way?

 A: The top comment right now pretty much answered the question...I'm just going to try and help you go through the process of visualizing it.
Lets name the two red circled changes in height. The one further back is $f_xdx$, and the one closer to us is $f_ydy$. We want to convince ourselves that $dz=f_xdx+f_ydy$
Start by imagining that $f(x,y)$ has zero slope in the $y$ direction. That is, $f_y=0$, and $dy$ doesn't change the height of the function at all.
Then, the tangent plane would only be inclining upwards in the $x$ direction. In this case, it shouldn't be too hard to see that $dz$ would simply be equal to the circled change in height further back. That is, $dz=f_xdx$.
The reason for this is because the points $(x,y)$ and $(x,y+dy)$ would have the same height, and the plane would only be inclining in the $x$ direction. So, $(x+dx,y)$ and $(x+dx,y+dy)$ would be at equal heights as well.
Now, lets add a slope in the $y$ direction. Imagine slowly inclining the plane so that it has a slope in the $y$ direction as well, while keeping the same slope $f_x$ in the $x$ direction.
As you do, the line connecting $(x,y)$ and $(x+dx,y)$ won't change, since $f_x$ didn't change, and thus the height of $(x+dx,y)$ won't change.
Additionally (and this is the crucial step), the line connecting $(x,y+dy)$ and $(x+dx,y+dy)$ must have the same slope $f_x$ in the $x$ direction as before, since we're not changing the slope in the $x$ direction, only the slope in the $y$ direction.
The only way to make sure of this is if the points $(x,y+dy)$ and $(x+dx,y+dy)$ rise up by the same amount. They will each rise up by $f_ydy$.
Before inclining the plane, the point $(x+dx,y+dy)$ was at a height of $f_xdx$. Now, it will rise by $f_ydy$, just like $(x,y+dy)$ will.
And thus, $dz=f_xdx+f_ydy$!!
Hope that made sense! Let me know if I should clear up any part of this!
