Find the equation of the normal to the curve \({x^3}{y^3} - xy = 0\) at the point (1, 1).

\({x^3}{y^3} - xy = 0\)

\(3{x^2}{y^3} + 3{x^3}{y^2}y' - y - xy' = 0\)

Note: Award A1 for correctly differentiating each term.

\(x = 1,{\text{ }}y = 1\)     \(3 + 3y' - 1 - y' = 0\)

     \(2y' = - 2\)

     \(y' = - 1\)     (M1)A1

gradient of normal = 1     (A1)

equation of the normal \(y - 1 = x - 1\)     A1     N2

\(y = x\)

Note: Award A2R5 for correct answer and correct justification.

[7 marks]

This implicit differentiation question was well answered by most candidates with many achieving full marks. Some candidates made algebraic errors which prevented them from scoring well in this question.

Other candidates realised that the equation of the curve could be simplified although the simplification was seldom justified.