Meaning of derivative of a function with respect to another function In chain rule we use,$\frac{dy}{dx}=\frac{dy}{dz} \times \frac{dz}{dx}$. My question is what is meant by $\frac{dy}{dz}$.When we find derivative,we find it with respect to an independent vairable meaning if $x$ changes by some amount,how much does $f(x)$ changes.But how much $g(f(x))$ changes by a change in $f(x)$ doesn't seem to make sense to me.Also derivatives are all about slopes of a graph.Now we will we draw the graph of $g(f(x))$ vs $f(x)$ since we can't place $f(x)$ on real axis as it can't achieve all real numbers for it being a dependent variable.So could someone please shed some light on what we are actually doing?

Consider this example: let's say we have a variable $V$ representing volume, and another variable $T$ representing temperature. Now the rate of change of volume with respect to temperature is given by $\frac{dV}{dT}$. If we further suppose that $T$ depends on another variable $t$ representing time, then the rate of change of volume with respect to time can be found by $$\frac{dV}{dt}=\frac{dV}{dT}\cdot \frac{dT}{dt}$$
