find the derivative of ( f(x) ).\n\n( f(x)=e^{8 x}+e^{x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{8x})

Use the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime). Let (u = 8x), then (u^\prime=8). So ((e^{8x})^\prime=e^{8x}\cdot8 = 8e^{8x}).

Step2: Differentiate (e^{x})

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

Step3: Sum the derivatives

Since (f(x)=e^{8x}+e^{x}), then (f^\prime(x)=(e^{8x})^\prime+(e^{x})^\prime).

Answer:

(8e^{8x}+e^{x})