if ( f(x)=\frac{4 x + 8}{3 x + 4} ), find:\n( f^{prime}(x)= )\n( f^{prime}(4)= )\nquestion help: video…

Explanation:

Step1: Apply the quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u = 4x + 8), so (u^\prime=4); (v = 3x + 4), so (v^\prime = 3). [ \begin{align*} f^\prime(x)&=\frac{(4)(3x + 4)-(4x + 8)(3)}{(3x + 4)^{2}}\ \end{align*} ]

Step2: Simplify the numerator

Expand ((4)(3x + 4)=12x+16) and ((4x + 8)(3)=12x + 24). [ \begin{align*} f^\prime(x)&=\frac{12x+16-(12x + 24)}{(3x + 4)^{2}}\ &=\frac{12x+16 - 12x-24}{(3x + 4)^{2}}\ &=\frac{- 8}{(3x + 4)^{2}} \end{align*} ]

Step3: Find (f^\prime(4))

Substitute (x = 4) into (f^\prime(x)). [ \begin{align*} f^\prime(4)&=\frac{-8}{(3\times4 + 4)^{2}}\ &=\frac{-8}{(12 + 4)^{2}}\ &=\frac{-8}{256}\ &=-\frac{1}{32} \end{align*} ]

Answer:

(f^\prime(x)=\frac{-8}{(3x + 4)^{2}}) (f^\prime(4)=-\frac{1}{32})