Calculus + Induction

Here’s the question we considered on Thursday. Now that you’ve got a solution for parts 1 and 2, complete part 3 as a homework assignment due on Tuesday, the 25th of October.

The function $f$is defined by $f(x) = e^x \sin x$.

1. Show that $f”(x) = 2e^x \sin \left(x+\frac{\pi}{2}\right)$.
2. Obtain a similar expression for $f^{(4)}(x)$.
3. Suggest an expression for $f^{(2n)}(x), n \in \mathbb{Z}^+$, and prove your conjecture using mathematical induction.