Question

find the derivative of ( f(x) ).

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

( 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:

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