find the derivative of ( f(x) ).\n\n( f(x)=e^{-7 x}+e^{-6 x} )\n\n( f^{prime}(x)= )

Explanation:

Step1: Differentiate (e^{-7x})

Using the chain rule ((e^{u})^\prime = e^{u}\cdot u^\prime), where (u = -7x) and (u^\prime=-7). So ((e^{-7x})^\prime=e^{-7x}\cdot(-7)=-7e^{-7x})

Step2: Differentiate (e^{-6x})

Using the chain rule ((e^{u})^\prime = e^{u}\cdot u^\prime), where (u = -6x) and (u^\prime=-6). So ((e^{-6x})^\prime=e^{-6x}\cdot(-6)=-6e^{-6x})

Step3: Combine the derivatives

Since (f(x)=e^{-7x}+e^{-6x}), then (f^\prime(x)=(e^{-7x})^\prime+(e^{-6x})^\prime)

Answer:

(f^\prime(x)=-7e^{-7x}-6e^{-6x})