Chapter 2.7, Problem 16E

To determine

To Sketch: The graph of $f\left(x\right)$ and then $f^{\prime }\left(x\right)$ below it.

Answer to Problem 16E

The formula for $f^{\prime }\left(x\right)$ from its graph is, $f^{\prime }\left(x\right)=\frac{1}{x}\text{ for all }x>0$.

Explanation of Solution

The given function is $f\left(x\right)=\ln x$.

From the graph of $f\left(x\right)=\ln x$, it is clear that the slope is always positive.

Thus, the derivative of the graph is always positive.

Draw the tangent to the graph at $x=1$ and the slope of the graph at $x=1$ is computed as follows,

$\begin{array}{c}m=\frac{0-1}{-1-0}\\ =\frac{-1}{-1}\\ =1\end{array}$

Recall the fact that, the slope of the function at the point $x=a$ is equal to the derivative of the function at that point.

Therefore, $f^{\prime }\left(x\right)=1$ at $x=1$.

Thus, the graph of the function $f^{\prime }\left(x\right)$ passes through the point $\left(1,\text{ 1}\right)$.

From the point A to left, the slope decreases rapidly. Thus, the graph of the function $f^{\prime }\left(x\right)$ in this section will decrease rapidly.

From the point A to right, the slope is increasing slowly. Thus, the graph of the function $f^{\prime }\left(x\right)$ in this section will decrease slowly.

Graph:

Use the online graphing calculator to draw the graph of $f\left(x\right)=\ln x$ and use the above information to trace the graph of $f^{\prime }\left(x\right)$ as shown below in Figure 1.

Observation:

From Figure 1, it is observed that the graph of the derivative function seems to be $\frac{1}{x}$ for $x>0$.

Thus, the derivative formula is, $f^{\prime }\left(x\right)=\frac{1}{x}\text{ for all }x>0$.