Question

Calculate y' when xy^4+x^2y=x+3y

Answer 1

All you need to do to solve such kind of problems is feel comfortable with implicit differentiation.

Tools you need:
- The Chain Rule (One tool to rule them all)
- The Product Rule of differentiation
- Learn to treat y as a function of x, that will make the above seem more intuitive:

\[\Large xy(x)^4+x^2y(x)=x+3y(x) \]

Answer 2

Maybe even a bit better written:

\[ \Large x\left(y(x)\right)^4+x^2y(x)=x+3y(x)\]

This way basically - a bit vaguely speaking - everything is written in terms of \(x\). So the use of the chain rule and product rule seems more obvious to me and some. Apply these two rules and solve for dy/dx