find the derivative of ( f(x) ).\n\n( f(x)=8 e^{-2 x}+7 e^{-6 x} )\n\n( f^{prime}(x)= )

Answer:

(-16e^{-2x}-42e^{-6x})

Explanation:

Step1: Differentiate (8e^{-2x})

Use the chain rule ((e^{u})^\prime = e^{u}\cdot u^\prime). Let (u = -2x), then (u^\prime=-2). So ((8e^{-2x})^\prime=8\times e^{-2x}\times(-2)=-16e^{-2x})

Step2: Differentiate (7e^{-6x})

Use the chain rule. Let (u = -6x), then (u^\prime=-6). So ((7e^{-6x})^\prime=7\times e^{-6x}\times(-6)=-42e^{-6x})

Step3: Sum the derivatives

(f^\prime(x)=(8e^{-2x}+7e^{-6x})^\prime=(8e^{-2x})^\prime+(7e^{-6x})^\prime=-16e^{-2x}-42e^{-6x})