Question

find the slope of the following curve at x = 6.
y = \frac{1}{x - 2}
the slope of the given curve at x = 6 is
(simplify your answer.)

Explanation:

Step1: Differentiate the function

We have $y=\frac{1}{x - 2}=(x - 2)^{-1}$. Using the power - rule for differentiation $\frac{d}{dx}(u^n)=nu^{n - 1}\frac{du}{dx}$, where $u=x - 2$ and $n=-1$. Then $y^\prime=-1\times(x - 2)^{-2}\times1=-\frac{1}{(x - 2)^2}$.

Step2: Evaluate the derivative at $x = 6$

Substitute $x = 6$ into $y^\prime$. We get $y^\prime|_{x = 6}=-\frac{1}{(6 - 2)^2}=-\frac{1}{16}$.

Answer:

$-\frac{1}{16}$