Answer
$f'(x) =2e^{2x}$
$f'(x) =4e^{-x}$
$f''(x) =8e^{-x}$
$f^{(4)}(x)=16e^{2x}$
$\cdots$
$f^{(n)}(x)=2^{n}e^{2x}$
Work Step by Step
$f'''(x)=\displaystyle \frac{d}{dx}\left[f''(x)\right]$
$f^{(4)}(x)=\displaystyle \frac{d}{dx}\left[f'''(x)\right]$
$\displaystyle \quad f^{(5)}(x)=\frac{d}{dx}\left[f^{(4)}(x)\right]$
and so on, assuming all these derivatives exist.
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$f(x)=e^{2x}$
$f'(x)=e^{2x}(2)=2e^{2x}$
$f'(x)=2e^{2x}(2)=4e^{-x}$
$f''(x)=4e^{2x}(2)=8e^{-x}$
$f^{(4)}(x)=16e^{2x}$
$\cdots$
$f^{(n)}(x)=2^{n}e^{2x}$
