Question

If f(x)=e^2x, find a formula for f^(n)(x)

Answer 1

Assuming
`f^((n))` denotes the
`n` derivative of
`f`, then notice that


`f(x)=e^(2x)`

`\implies f'(x)=2e^(2x)=2f(x)`

`\implies f''(x)=2f'(x)`

`\implies f'''(x)=2f''(x)`

`\implies\cdots\implies f^((n+1))(x)=2f^((n))(x)`

`\implies f^((n))(x)=2^nf(x)=2^ne^(2x)`