Question

find f(x) and simplify. f(x) = (3x - 4)/(x² + 7) which of the following shows the correct application of the quotient rule? a. (x² + 7)(3) - (3x - 4)(2x)/(3x - 4)² b. (3x - 4)(2x) - (x² + 7)(3)/(x² + 7)² c. (x² + 7)(3) - (3x - 4)(2x)/(x² + 7)² d. (3x - 4)(2x) - (x² + 7)(3)/(3x - 4)²

Explanation:

Step1: Recall quotient - rule formula

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = x^{2}+7$ and $v=3x - 4$.

Step2: Find $u^\prime$ and $v^\prime$

Differentiate $u = x^{2}+7$ with respect to $x$: $u^\prime = 2x$. Differentiate $v = 3x - 4$ with respect to $x$: $v^\prime=3$.

Step3: Apply the quotient - rule

$f^\prime(x)=\frac{(2x)(3x - 4)-(x^{2}+7)(3)}{(3x - 4)^{2}}$.

Answer:

B. $\frac{(2x)(3x - 4)-(x^{2}+7)(3)}{(3x - 4)^{2}}$