| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find second derivative d²y/dx² |
| Difficulty | Standard +0.8 This is a Further Maths implicit differentiation question requiring two applications of the technique. Part (i) is standard FP1 fare, but part (ii) requires differentiating the first derivative expression implicitly again, which involves careful application of quotient/product rules and substitution. The algebraic manipulation is non-trivial but follows established procedures for second derivatives of implicit functions. |
| Spec | 1.07s Parametric and implicit differentiation |
7 The curve $C$ has equation
$$x y + ( x + y ) ^ { 5 } = 1$$
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }$ at the point $A ( 1,0 )$ on $C$.\\
(ii) Find the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at $A$.
\hfill \mbox{\textit{CAIE FP1 2004 Q7 [8]}}