Question

if ( f(x)=\frac{4 x + 8}{3 x + 4} ), find:
( f^{prime}(x)= )
( f^{prime}(4)= )
question help: video message instructor
submit question jump to answer

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 \(v^\prime = 3\).

$$
LATEXBLOCK0
$$

Step2: Simplify the numerator

Expand \((4)(3x + 4)=12x+16\) and \((4x + 8)(3)=12x + 24\).

$$
LATEXBLOCK1
$$

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

Substitute \(x = 4\) into \(f^\prime(x)\).

$$
LATEXBLOCK2
$$

Answer:

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