Standard +0.3 This is a standard implicit differentiation question requiring two applications of the technique. Finding dy/dx involves routine application of the chain and product rules, and the second derivative requires careful but methodical differentiation of the first result. While it requires multiple steps and careful algebra, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average for Further Maths students.
4 Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { 3 }\) at the point \(A ( 1 , - 2 )\) on the curve with equation
$$y ^ { 3 } - 3 x ^ { 2 } y + 2 = 0$$
and find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
4 Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { 3 }$ at the point $A ( 1 , - 2 )$ on the curve with equation
$$y ^ { 3 } - 3 x ^ { 2 } y + 2 = 0$$
and find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.
\hfill \mbox{\textit{CAIE FP1 2013 Q4 [8]}}