What is involved with deriving demand?

We differentiate utility twice; the cross partial gives the demand curve

We solve one constrained maximization problem for each point on the demand curve

The demand curve for a given indifference curve is given by the intersections of that indifference curve with all possible budget constraints

The demand curve for a given budget constraint is given by the intersections of that budget constraint with all possible indifference curves

Simply differentiate the indifference curve