Question

let (f) be a differentiable function such that (f(8)=2) and (f(8)=5). if (g) is the function defined by (g(x)=\frac{f(x)}{x}), what is the value of (g(8))?

Explanation:

Step1: Apply quotient - rule for differentiation

The quotient - rule states that if $g(x)=\frac{f(x)}{x}$, then $g'(x)=\frac{f'(x)\cdot x - f(x)\cdot1}{x^{2}}$.

Step2: Substitute $x = 8$

We know that $f(8) = 2$ and $f'(8)=5$. Substitute these values into the formula for $g'(x)$.
$g'(8)=\frac{f'(8)\cdot8 - f(8)\cdot1}{8^{2}}$.
$g'(8)=\frac{5\times8 - 2\times1}{64}$.
$g'(8)=\frac{40 - 2}{64}=\frac{38}{64}=\frac{19}{32}$.

Answer:

$\frac{19}{32}$