find the derivative of ( f(x) ).\n\n( f(x)=e^{4 x}+e^{x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{4x})

Use the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime). Let (u = 4x), then (u^\prime=4). So ((e^{4x})^\prime=e^{4x}\cdot4 = 4e^{4x})

Step2: Differentiate (e^{x})

For (y = e^{x}), its derivative is (e^{x}) according to the formula ((e^{x})^\prime=e^{x})

Step3: Sum the derivatives

Since (f(x)=e^{4x}+e^{x}), by the sum rule ((u + v)^\prime=u^\prime+v^\prime), we have (f^\prime(x)=(e^{4x})^\prime+(e^{x})^\prime)

Answer:

(4e^{4x}+e^{x})