Finding the Derivative of xln(x) Video

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Finding the Derivative of 1/cos(x)

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 Steps to Solve
  • 2:40 Checking Your Work…
  • 5:05 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Timeline
Autoplay
Autoplay
Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will see how to use the product rule for derivatives to find the derivative of xln(x). Once we have found our derivative, we will look at how we can check our work and know we did everything correctly.

Steps to Solve

In order to find the derivative of xln(x), we are going to be using the product rule for derivatives. The product rule for derivatives states that to take the derivative of a product of functions, we multiply the derivative of the first function times the second function, and add it to the derivative of the second function multiplied by the first function. The following equation shows this in symbol form:


derxlnx1


We can use this rule to find the derivative of xln(x) because this is a product of the functions f(x) = x and g(x) = ln(x). There are a couple more facts that we will need to know in order to find this derivative.

  • The derivative of x is 1. This comes from the fact that the derivative of xn is nxn-1. If we look at x as x1, we have that the derivative of x1 is 1 * x1 - 1 = x0 = 1.
  • The derivative of ln(x) is 1/x. This can be observed by looking at the slope of the tangent line at various values of x because the derivative of a function at a point is the slope of the tangent line at that point. Observe the following chart with the given slopes of the tangent line of ln(x) at various values of x:

x Slope of the tangent line of ln(x) at x
1 1
2 1/2
3 1/3
4 1/4
5 1/5

If we continue this pattern, we see that the slope of the tangent line of ln(x) at a given value of x is 1/x, so the derivative of ln(x) is 1/x.

These two facts, along with the product rule, will allow us to find the derivative of xln(x).

As we said, xln(x) is the product of f(x) = x and g(x) = ln(x). In order to use the product rule to find the derivative, we need to know f ' (x) and g ' (x).

From the facts, the derivative of x is 1, so f ' (x) = 1.

Also from the facts, the derivative of ln(x) is 1/x, so g ' (x) = 1/x.

Now we simply plug into the product rule for derivatives and simplify.


derxlnx2


We see that the derivative of xln(x) is ln(x) + 1.

Checking Your Work with Integrals

After finding your derivative, you may want to have a way to know for sure that you found it correctly. Thankfully, we have a tool to check our work, and that tool is integrals! Integrals are a subject that are studied after derivatives are mastered. If you haven't studied them already, when you do you will learn that they have a close relationship with derivatives. In fact, integrals are called anti-derivatives, and they are related to derivatives in that if a is the derivative of b, then the integral of a is b + C, where C is a constant.


derxlnx3


To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account
Support