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

$f(x)=e^{2x}$
$\implies f'(x)=2e^{2x}=2f(x)$
$\implies f''(x)=2f'(x)$
$\implies f'''(x)=2f''(x)$
$\implies\cdots\implies f^{(n+1)}(x)=2f^{(n)}(x)$
$\implies f^{(n)}(x)=2^nf(x)=2^ne^{2x}$