Standard +0.8 This Further Maths question requires implicit differentiation of a non-standard equation involving (dy/dx)³, then differentiating again to find d²y/dx². Students must first solve a cubic equation, then carefully apply the chain rule and product rule in the second differentiation. The unusual form and multi-step nature make it moderately challenging but still within standard FP1 scope.
1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 1\) and
$$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$
Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
1 The variables $x$ and $y$ are such that $y = - 1$ when $x = 1$ and
$$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$
Find the values of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ when $x = 1$.
\hfill \mbox{\textit{CAIE FP1 2010 Q1 [5]}}