find the derivative of ( f(x) ).\n\n( f(x)=5 e^{-6 x}-4 e^{-x} )\n\n( f^{prime}(x)= )

Answer:

(-30e^{-6x}+4e^{-x})

Explanation:

Step1: Differentiate (5e^{-6x})

Use the chain rule ((e^{u})^\prime = e^{u}\cdot u^\prime). Let (u = -6x), then (u^\prime=-6). So ((5e^{-6x})^\prime=5e^{-6x}\cdot(-6)=-30e^{-6x})

Step2: Differentiate (-4e^{-x})

Let (u=-x), then (u^\prime = -1). So ((-4e^{-x})^\prime=-4e^{-x}\cdot(-1) = 4e^{-x})

Step3: Combine the derivatives

(f^\prime(x)=(5e^{-6x}-4e^{-x})^\prime=(5e^{-6x})^\prime-(4e^{-x})^\prime=-30e^{-6x}+4e^{-x})