Answer
$2$
Work Step by Step
We are given the function:
$f(x)=\tan x$
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 $x=\dfrac{\pi}{4}$, consider intervals $[x_1,x_0],[x_0,x_1]$ for $x_1$ close to $x_0$, where $x_0=\dfrac{\pi}{4}$:
$\left[\dfrac{\pi}{3.9},\dfrac{\pi}{4}\right]$: $\dfrac{\Delta f}{\Delta x}=\dfrac{f\left(\dfrac{\pi}{4.9}\right)-f\left(\dfrac{\pi}{4}\right)}{\dfrac{\pi}{3.9}-\dfrac{\pi}{4}}=\dfrac{\tan\dfrac{\pi}{3.9}-\tan\dfrac{\pi}{4}}{\dfrac{\pi}{3.9}-\dfrac{\pi}{4}}\approx 2.0413863$
See table
From the table we find that the instantaneous rate of of change at $x=\dfrac{\pi}{4}$ is approximately $2$.
