This is known as implicit differentiation.
Let's say we have
The derivative of with respect to t is
In this case, we have
and the derivative of which (with respect to t) is
In this case, however, y depends on t so doesn't equate to 1 like in the above example. So we must keep it.
By the same token,
We are taking the derivative of the above with respect to t, which is equal to
So basically we're taking the derivative of the left side with respect to t, and of the right side with respect to t. But since the left side is depdent on t and we dont have an equivilent expression for it, we leave it as