Standard +0.8 This is a Further Maths implicit differentiation question requiring both first and second derivatives. While the first derivative is straightforward product rule application, finding d²y/dx² implicitly requires careful differentiation of dy/dx (which itself contains dy/dx terms), then substitution of known values. This is more demanding than standard C3/C4 implicit differentiation and requires systematic algebraic manipulation, placing it moderately above average difficulty.
4 The curve \(C\) has equation
$$2 x y ^ { 2 } + 3 x ^ { 2 } y = 1$$
Show that, at the point \(A ( - 1,1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - 4\).
Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
4 The curve $C$ has equation
$$2 x y ^ { 2 } + 3 x ^ { 2 } y = 1$$
Show that, at the point $A ( - 1,1 )$ on $C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - 4$.
Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.
\hfill \mbox{\textit{CAIE FP1 2011 Q4 [8]}}