find the slope of the following curve at x = 7. y = 1/(x - 5) the slope of the given curve at x = 7 is -1/4…

Explanation:

Step1: Rewrite the function

Rewrite $y = \frac{1}{x - 5}$ as $y=(x - 5)^{-1}$.

Step2: Apply the power - rule for differentiation

The power - rule states that if $y = u^n$, then $y^\prime=nu^{n - 1}u^\prime$. Here $u=x - 5$, $n=-1$, and $u^\prime = 1$. So $y^\prime=-1\times(x - 5)^{-2}\times1=-\frac{1}{(x - 5)^2}$.

Step3: Evaluate the derivative at $x = 7$

Substitute $x = 7$ into $y^\prime$. We get $y^\prime\big|_{x = 7}=-\frac{1}{(7 - 5)^2}=-\frac{1}{4}$.

Answer:

$-\frac{1}{4}$