if ( f(x)=(3 x + 2)^{-3} ),\nfind ( f^{prime}(x) ).\nthen ( f^{prime}(x)=)\nfind ( f^{prime}(4) ).\nthen (…

Explanation:

Step1: Apply the chain rule

Let (u = 3x+2), then (y = u^{-3}). The chain rule states that (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). First, find (\frac{dy}{du}): If (y = u^{-3}), then (\frac{dy}{du}=-3u^{-4}). Next, find (\frac{du}{dx}): If (u = 3x + 2), then (\frac{du}{dx}=3). So, (\frac{dy}{dx}=-3u^{-4}\cdot3=-9(3x + 2)^{-4}).

Step2: Evaluate (f^{\prime}(4))

Substitute (x = 4) into (f^{\prime}(x)): (f^{\prime}(4)=-9(3\times4 + 2)^{-4}) (=-9(12 + 2)^{-4}) (=-9\times14^{-4}) (=-\frac{9}{14^{4}}) (=-\frac{9}{38416})

Answer:

(f^{\prime}(x)=-9(3x + 2)^{-4}) (f^{\prime}(4)=-\frac{9}{38416})