I have been stuck on this problem for a while and I need help!
A rectangle exists in the coordinate plane so that one of its sides is a segment on the $y$-axis and the adjacent side is a segment on the $x$-axis. The upper-right corner of the rectangle lies on the curve $x^3-2xy^2+y^3+1=0$, and is moving (causing the sides of the rectangle to change their lengths). How fast is the area of the rectangle changing at the instant the corner passes through the point $(2,3)$ if the corner is moving so that $\frac{dx}{dt}=1$ unit per second?
Hint: Draw a picture. Even if you don't know what the curve looks like, draw a generic curve so you can label your rectangle!
Hint: You are looking for the rate of change of the area, but you should represent it with an expression of given variables instead of calling it $A$
Hint: This is going to involve implicit differentiation
Hint: Like some types of shadow problems, you'll have to solve for one rate before you can find the rate you want