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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 5: Derivative as a limit

# Derivative as a limit: numerical

## Problem

Jude wants to find the derivative of $h\left(x\right)=12\mathrm{ln}\left(x\right)$ at the point $x=4$.
His table below shows the average rate of change of $h$ over the intervals $\left[x,4\right]$ or $\left[4,x\right]$ for $x$-values that get increasingly closer to $4$:
$x$IntervalAverage rate of change, $\frac{h\left(x\right)-h\left(4\right)}{x-4}$
$3.9$$\left[3.9,4\right]$$3.0381$
$3.99$$\left[3.99,4\right]$$3.0038$
$3.999$$\left[3.999,4\right]$$3.0004$
$4.001$$\left[4,4.001\right]$$2.9996$
$4.01$$\left[4,4.01\right]$$2.9963$
$4.1$$\left[4,4.1\right]$$2.9631$
From the table, what does the derivative of $h\left(x\right)=12\mathrm{ln}\left(x\right)$ at $x=4$ appear to be?