I saw this in Quora: $$ \frac{d}{dt}\, \frac{dy}{dx} = \left(\frac{d}{dx}\, \frac{dy}{dx}\right) \frac{dx}{dt}. $$
I tried understanding this equation but I couldn't. I think it's not valid after searching wikipedia and multiple text books.
I know that there is composition going on but I can't really figure out what is a function in this expression.
I understand all the other notations but this one boggles me.
We can write the definition of the chain rule as $$\frac{dz}{dt} = \frac{dz}{dx} \cdot \frac{dx}{dt}.$$ Now, take $z={dy\over dx}$ and plug it in the previous formula: $${d\over dt}{dy\over dx}=\left({d\over dx}{dy\over dx}\right){dx\over dt}\quad\text{(1)}$$ You get the expression of your question.
In particular, $\text{(1)}$ is not a definition of the chain rule, but an application of it.