# What can go wrong with applying chain rule to angular velocity of circular motion?

The reason is that $$r(t)$$ and $$\phi(t)$$ are your independent variables, so you are not allowed to use the chain rule for them, because $$\phi$$ does not depend on $$r$$. An analogy would be the following: $$\begin{equation} \frac{d}{dx}x=1 \end{equation}$$ However, if I introduce another independent variable $$y$$, it would be of course wrong to write $$\begin{equation} \frac{d}{dx}x=\frac{dy}{dx}\frac{d}{dy}x=0 \end{equation}$$

The meaning of pushing an intermediate variable is to relate the dependencies of change. So, the expression

$$\frac{d\phi}{dt} = \frac{d \phi}{dr} \frac{dr}{dt}$$

Assumes that the angle $$(\phi)$$ can be written as some function of the radius from the origin i.e: $$\phi(r(t))$$ but is that really possible in this case? We can say that there is no 'differentiable' map from $$r \to \phi$$.

I mean think about it, how would you construct a function which associates the radius , a fixed variable, with the angle which changes with time? It would not even be a function because you would have to span all the angles with a single radius value.

Though $$\frac{\mathrm{d}r}{\mathrm{d}t}$$ is $$0$$, $$\frac{\mathrm{d}\phi}{\mathrm{d}r}$$ also tends to infinity. So this is a zero into infinity form and its limit will result in a finite quantity.