find a formula for (f^{(n)}(x)) if (f(x)=e^{2x}).\n(f^{(n)}(x)=)

Explanation:

Step1: Recall derivative of exponential

The derivative of $y = e^{ax}$ is $y'=ae^{ax}$. For $f(x)=e^{2x}$, $f'(x) = 2e^{2x}$.

Step2: Find second - derivative

Differentiate $f'(x)=2e^{2x}$ again. Using the rule, $f''(x)=\frac{d}{dx}(2e^{2x}) = 2\times2e^{2x}=2^{2}e^{2x}$.

Step3: Generalize for $n$ - th derivative

By induction, if we keep differentiating, the $n$-th derivative $f^{(n)}(x)$ of $f(x)=e^{2x}$ will be $2^{n}e^{2x}$.

Answer:

$2^{n}e^{2x}$