- Given that
$$y = \mathrm { e } ^ { 2 x } \sinh x$$
prove by induction that for \(n \in \mathbb { N }\)
$$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = \mathrm { e } ^ { 2 x } \left( \frac { 3 ^ { n } + 1 } { 2 } \sinh x + \frac { 3 ^ { n } - 1 } { 2 } \cosh x \right)$$