find the slope of the curve at the point indicated y = (x - 5)/(x + 5), x = - 3 when x = - 3, the slope is…

Explanation:

Step1: Apply quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = x - 5$, so $u^\prime=1$; $v=x + 5$, so $v^\prime = 1$. Then $y^\prime=\frac{1\times(x + 5)-(x - 5)\times1}{(x + 5)^{2}}$.

Step2: Simplify the derivative

Expand the numerator: $y^\prime=\frac{x + 5-(x - 5)}{(x + 5)^{2}}=\frac{x + 5 - x+5}{(x + 5)^{2}}=\frac{10}{(x + 5)^{2}}$.

Step3: Evaluate the derivative at $x=-3$

Substitute $x=-3$ into $y^\prime$: $y^\prime|_{x = - 3}=\frac{10}{(-3 + 5)^{2}}=\frac{10}{4}=\frac{5}{2}$.

Answer:

$\frac{5}{2}$