How Do Differentials Apply to Implicit Functions in Calculus?

[Timothy S]

I have recently come across the use of differentials in visualizing and thinking about calculus. In this method, one thinks of dx/dy as an actual fraction of infinity small yet real numbers. How is it possible to apply this to implicit functions?

 

[pwsnafu]

infinity small yet real numbers

Contradiction in terms. Every real number is finite by definition. What you can do is consider non-real numbers that satisfy $0<\epsilon<r$ for every real number r. This is the core of infinitesimals.