Quadratic sine derivative pattern

Let \(\mathrm{f}(x)=x^2\sin x\).

1. Using the definition \(\mathrm{g}'(x)=\lim_{h\to0}\dfrac{\mathrm{g}(x+h)-\mathrm{g}(x)}{h}\), show from first principles that the derivative of \(\mathrm{g}(x)=x^2\) is \(2x\).[4]
2. Prove by induction that, for every positive integer \(n\),
   \[\mathrm{f}^{(n)}(x)=[x^2-n(n-1)]\sin\left(x+\frac{n\pi}{2}\right)-2nx\cos\left(x+\frac{n\pi}{2}\right).\][9]