Answer
$-0.0625$
Work Step by Step
We are given the function:
$y(x)=\dfrac{1}{x+2}$
The average rate of change on $[x_0,x_1]$ is:
$\dfrac{\Delta y}{\Delta x}=\dfrac{y(x_1)-y(x_0)}{x_1-x_0}$
The instantaneous rate of change is the limit of the average rate of change.
In order to estimate the instantaneous rate of of change at $x=2$, consider intervals $[x_1,x_0],[x_0,x_1]$ for $x_1$ close to $x_0$, where $x_0=2$:
$[1.9,2]$: $\dfrac{\Delta y}{\Delta x}=\dfrac{y(1.9)-y(2)}{1.9-2}=\dfrac{\dfrac{1}{1.9+2}-\dfrac{1}{2+2}}{-0.1}\approx -0.064102564$
See table
From the table we find that the instantaneous rate of of change at $x=2$ is approximately $-0.0625$.
