Question

find the derivative of ( f(x) ).

( f(x)=5 e^{-6 x}-4 e^{-x} )

( f^{prime}(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

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

Answer:

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