Chapter 2.1, Problem 24E

To determine

The derivative of the function.

Answer to Problem 24E

The derivative of the function is $y=\ln x$ .

Explanation of Solution

Given information:

The function is $f\left(x\right)=x\ln x-x$ and the graph of the function is as shown below.

  

And the options are (a) $y=e^{x-1}$ , (b) $y=\ln x$ , (c) $y=-\ln x$ , (d) $y=\ln x-1$ , (e) $y=3x-1$ .

Concept used:

If the function is decreasing from left to right then the value of derivative should be in negative $y$ axis and if the function is increasing from left to right then the value of derivative should be in positive $y$ axis.

Calculation:

Graph of the function $y=e^{x-1}$ is as shown below.

  

The function is first decreasing then increasing so the derivative should have negative value first then positive value.

But in this case the function has positive value first then negative value from left to right.

So it cannot be derivative of the function.

Graph of the function $y=\ln x$ is as shown below.

  

The function is first decreasing then increasing so the derivative should have negative value first then positive value.

In this case the function has negative value first then positive value from left to right.

So it is derivative of the function.

Graph of the function $y=-\ln x$ is as shown below.

  

The function is first decreasing then increasing so the derivative should have negative value first then positive value.

But in this case the function has positive value first then negative value from left to right.

So it cannot be derivative of the function.

Graph of the function $y=\ln x-1$ is as shown below.

  

The function is first decreasing then increasing so the derivative should have negative value first then positive value.

In this case the function has negative value first then positive value from left to right also the tangent is at $x=0$

So it cannot be derivative of the function.

Graph of the function $y=3x-1$ is as shown below.

  

The function is first decreasing then increasing so the derivative should have negative value first then positive value.

In this case the function has negative value first then positive value from left to right also it has a straight line.

So it cannot be derivative of the function.

Conclusion: So the only function having its derivative is $y=\ln x$ .