Question

find the derivative of ( f(x) ).

( f(x)=8 e^{-2 x}+7 e^{-6 x} )

( f^{prime}(x)= )

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

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

Answer:

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