Question

what does dy/du; du/dx; and dy/dx mean in general terms

Answer 1

i think this is in reference to the chain rule

Answer 2

for example, if you have \(y=\sin(u)\) and \(u=x^2\) then \(y(x)=\sin(x^2)\) and to take the derivative you have
\[\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\] in my example above \(\frac{dy}{du}=\frac{d}{du}\sin(u)=\cos(u)\) and \(\frac{du}{dx}=\frac{d}{dx}x^2=2x\)
so
\[\frac{dy}{dx}=\cos(u)\times 2x=\cos(x^2)\times 2x\]

Answer 3

Basically, they are all relatives rates of changes. For instance, \(\dfrac{dy}{du}\) is the rate of change of \(y\) with respect to \(u\), and so on.