Challenging +1.2 This question combines differentiation with proof by induction. The differentiation part is routine (product rule applied repeatedly), and the induction follows a standard structure once the pattern is spotted. However, it requires careful algebraic manipulation across multiple derivatives and the ability to formulate and prove a general pattern, which elevates it above average difficulty for A-level but remains within standard Further Maths territory.
Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that
$$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3]
Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]
Given that $y = x \sin x$, find $\frac{d^2y}{dx^2}$ and $\frac{d^4y}{dx^4}$, simplifying your results as far as possible, and show that
$$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3]
Use induction to establish an expression for $\frac{d^{2n}y}{dx^{2n}}$, where $n$ is a positive integer. [5]
\hfill \mbox{\textit{CAIE FP1 2003 Q4 [8]}}