find the derivative of ( f(x) ).\n\n( f(x)=e^{x}+e^{3 x} )\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^x)

The derivative of (e^x) with respect to (x) is (e^x).

Step2: Differentiate (e^{3x})

Using the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime), where (u = 3x) and (u^\prime=3). So the derivative of (e^{3x}) is (e^{3x}\cdot3 = 3e^{3x}).

Step3: Sum the derivatives

(f^\prime(x)=(e^x)^\prime+(e^{3x})^\prime)

Answer:

(e^x + 3e^{3x})