Question

find the slope of the following curve at ( x = 7 ).
( y=\frac{1}{x - 4} )
the slope of the given curve at ( x = 7 ) is (simplify your answer.)

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:

\(-\frac{1}{9}\)