find the derivative of ( f(x) ).\n\n( f(x)=e^{6 x}+e^{x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{6x})

Use the chain rule ((e^{u})^\prime=e^{u}\cdot u^\prime). Let (u = 6x), then (u^\prime=6). So ((e^{6x})^\prime=e^{6x}\cdot6 = 6e^{6x}).

Step2: Differentiate (e^{x})

The derivative of (e^{x}) with respect to (x) is (e^{x}) (since for (y = e^{x}), (y^\prime=e^{x}) as the derivative of (e^{u}) with (u=x) and (u^\prime = 1), so ((e^{x})^\prime=e^{x}\cdot1=e^{x})).

Step3: Sum the derivatives

By the sum rule ((f(x)+g(x))^\prime=f^\prime(x)+g^\prime(x)), if (f(x)=e^{6x}) and (g(x)=e^{x}), then (f^\prime(x)+g^\prime(x)=6e^{6x}+e^{x}).

Answer:

(6e^{6x}+e^{x})