Question

find the derivative of ( f(x) ).

( f(x)=e^{3 x}+e^{-2 x} )

( f^{prime}(x)= )

Explanation:

Step1: Differentiate \(e^{3x}\)

Use the chain rule \((e^{u})^\prime = e^{u}\cdot u^\prime\). Let \(u = 3x\), then \(u^\prime=3\). So \((e^{3x})^\prime=e^{3x}\cdot3 = 3e^{3x}\)

Step2: Differentiate \(e^{-2x}\)

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

Step3: Combine the derivatives

Since \(f(x)=e^{3x}+e^{-2x}\), then \(f^\prime(x)=(e^{3x})^\prime+(e^{-2x})^\prime\)

Answer:

\(3e^{3x}-2e^{-2x}\)