I was looking at the following answer to a related rates problem, and it got me thinking: How is that the variable $t$ comes up in the bottom differential? If I multiply both sides by $dt$ and then divide by one of the remaining differentials I get the results of an implicit differentiation, but I was led to believe that one could not multiply and divide differentials so carelessly, and that doing so is a part of non-standard analysis. Is there a more standard way of doing this (maybe the Leibniz Chain Rule?) or am I just missing something entirely? I was always told to solve problems this way, and have... I just feel like my learning could have been a bit more rigid!
HINT:
You start with Pythagorean Theorem,
$$a^2 + b^2 = c^2$$
Then you take the derivative of both sides, and divide both sides by 2,
$$2a\frac{da}{dt} + 2b\frac{db}{dt} = 2c\frac{dc}{dt}$$ $$a\frac{da}{dt} + b\frac{db}{dt} = c\frac{dc}{dt}$$
In related rates problems, the variables of the problem are usually changing with respect to a common parameter, usually time. Thus, every variable in the problem becomes a function of time. In your example, the sides of the triangle $a$, $b$ and $c$ become functions of time $a(t)$, $b(t)$ and $c(t)$. This is where the $t$ in the bottom differential comes from. We know that
$$a(t)^2+b(t)^2=c(t)^2$$
When we differentiate with respect to $t$, the chain rule still applies and we obtain
$$2a(t)\frac{da}{dt}+2b(t)\frac{db}{dt}=2c(t)\frac{dc}{dt}$$ $$a(t)\frac{da}{dt}+b(t)\frac{db}{dt}=c(t)\frac{dc}{dt}$$
However, we usually don't write the variables explicitly as functions of time as I have done here; it is implied that they are functions of time as you have notated in your question.