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

Explanation:

Step1: Identify the outer and inner functions

Let ( u=-3x + 2 ), then ( f(x)=e^{u} ).

Step2: Differentiate the outer function

The derivative of ( y = e^{u} ) with respect to ( u ) is ( \frac{dy}{du}=e^{u} ).

Step3: Differentiate the inner function

The derivative of ( u=-3x + 2 ) with respect to ( x ) is ( \frac{du}{dx}=-3 ).

Step4: Apply the chain - rule

By the chain - rule ( \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx} ). Substitute ( \frac{dy}{du}=e^{u} ) and ( \frac{du}{dx}=-3 ) into the formula. Since ( u=-3x + 2 ), we have ( f^{\prime}(x)=e^{-3x + 2}\cdot(-3) ).

Answer:

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