Assuming [tex]f^{(n)}[/tex] denotes the [tex]n[/tex] derivative of [tex]f[/tex], then notice that
[tex]f(x)=e^{2x}[/tex]
[tex]\implies f'(x)=2e^{2x}=2f(x)[/tex]
[tex]\implies f''(x)=2f'(x)[/tex]
[tex]\implies f'''(x)=2f''(x)[/tex]
[tex]\implies\cdots\implies f^{(n+1)}(x)=2f^{(n)}(x)[/tex]
[tex]\implies f^{(n)}(x)=2^nf(x)=2^ne^{2x}[/tex]
