find the derivative of ( f(x) ).\n\n( f(x)=e^{3 x}+e^{-2 x} )\n\n( 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})