Answer
$0.866$
Work Step by Step
We are given the function:
$f(x)=\sin x$
The average rate of change on $[x_0,x_1]$ is:
$\dfrac{\Delta f}{\Delta x}=\dfrac{f(x_1)-f(x_0)}{x_1-x_0}$
The instantaneous rate of change is the limit of the average rate of change.
In order to estimate the instantaneous rate of change at $x=\dfrac{\pi}{6}$, consider intervals $[x_1,x_0],[x_0,x_1]$ for $x_1$ close to $x_0$, where $x_0=\dfrac{\pi}{6}$:
$\left[\dfrac{\pi}{5.9},\dfrac{\pi}{6}\right]$: $\dfrac{\Delta f}{\Delta x}=\dfrac{f\left(\dfrac{\pi}{5.9}\right)-f\left(\dfrac{\pi}{6}\right)}{\dfrac{\pi}{5.9}-\dfrac{\pi}{6}}=\dfrac{\sin\dfrac{\pi}{5.9}-\sin\dfrac{\pi}{6}}{\dfrac{\pi}{5.9}-\dfrac{\pi}{6}}\approx 0.86379541$
See table
From the table we find that the instantaneous rate of of change at $x=\dfrac{\pi}{6}$ is approximately $0.866$.
