Question

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

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