find the slope of the following curve at ( x = 7 ).\n( y=\frac{1}{x - 4} )\nthe slope of the given curve at…

Explanation:

Step1: Differentiate the function

Use the power rule ((x^n)^\prime = nx^{n - 1}). Rewrite (y=\frac{1}{x - 4}=(x - 4)^{-1}). By the chain rule ((u^n)^\prime=nu^{n - 1}\cdot u^\prime) (where (u=x - 4), (n=-1), (u^\prime = 1)), we have (y^\prime=-1\times(x - 4)^{-2}\times1=-\frac{1}{(x - 4)^2}).

Step2: Substitute (x = 7) into the derivative

Substitute (x = 7) into (y^\prime=-\frac{1}{(x - 4)^2}). We get (y^\prime\mid_{x = 7}=-\frac{1}{(7 - 4)^2}). Since ((7 - 4)^2=9), then (y^\prime\mid_{x = 7}=-\frac{1}{9}).

Answer:

(-\frac{1}{9})