Challenging +1.2 This is a structured induction proof requiring the product rule and chain rule to show the nth derivative has the stated form. While it involves higher derivatives and polynomial analysis, the induction framework is standard and the differentiation mechanics are routine for Further Maths students. The polynomial degree and leading coefficient verification adds mild complexity beyond basic induction proofs.
3 Prove by induction that, for all \(n \geqslant 1\),
$$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$
where \(\mathrm { P } _ { n } ( x )\) is a polynomial in \(x\) of degree \(n\) with the coefficient of \(x ^ { n }\) equal to \(2 ^ { n }\).
3 Prove by induction that, for all $n \geqslant 1$,
$$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$
where $\mathrm { P } _ { n } ( x )$ is a polynomial in $x$ of degree $n$ with the coefficient of $x ^ { n }$ equal to $2 ^ { n }$.
\hfill \mbox{\textit{CAIE FP1 2007 Q3 [6]}}