A function \(f\) has the rule \(f(x)=x\,e^{2x}\).
Use mathematical induction to prove that \(f^{(n)}(x)=\big{(}2^{n}x+n\,2^{n-1}\big{)}e^{2x}\) for \(n \in \mathbb{Z}^{+}\), where \(f^{(n)}(x)\) represents the \(n\)th derivative of \(f(x)\). That is, \(f(x)\) has been differentiated \(n\) times. (3 marks)
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