TITLE: What does $\frac{dy}{du}$ mean?
QUESTION [7 upvotes]: I understand that $dy/dx$ is the rate of change, and that it means "the rate of change of $y$ with respect to $x$", but when I see people use $dy/du$ I get confused ($u$ of course being any variable). What is meant by using $du$ or anything else but $dx$ in $dy/du$? When I look at a graph I can only see a $y$ and $x$ axis. 
Here is a video example of $dy/du$:
I'm new to calculus and I'm trying to understand the chain rule.

REPLY [2 votes]: If $dy/dx=6$ when $x=50$ and $y=17$, that means $y$ is changing $6$ times as fast as $x$ is changing at that point.
Leibniz in the 17th century and Euler in the 18th century thought of it like this: $dy$ is the infinitely small change in $y$ corresponing to and infinitely small change $dx$ in $x$.  The reason they have to be infinitely small is that the rates of change may be changing.
Leibniz is the one who introduced this notation in the first place.
If we were comfortable treating $x$ and $y$ as actual numbers, then we could say right away that when $x=50$ and $y=17$, then $dx/dy=1/6$.  But mathematicians have shied away from that approach because of logical difficulties, and in the 19th century invented more cumbersome ways to showed that if $dy/dx=6$ when $x=50$ and $y=17$, then $dx/dy=1/6$ at that point.  And those more cumbersome ways are a proof of the chain rule.  One way of viewing the chain rule is to say that $\dfrac{dy}{dy}\cdot\dfrac{du}{dx} =\dfrac{dy}{dx}$, so if $y$ is changing $2$ times as fast as $u$ and $u$ is changing $3$ times as fast as $x$, then $y$ is changing $6$ times as fast as $x$.  Again if we treat $dy$, $du$, and $dx$ as actual numbers, then this is merely a matter of canceling $du$.  But modern proofs of the chain rule are more involved.