Solving the Derivative of ln(sqrt x)

Solving the Derivative of ln(sqrt x)
Coming up next: Finding the Derivative of sec^2(x)

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

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 Steps to Solve
  • 2:40 Application
Add to Add to Add to

Want to watch this again later?

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

Log in or Sign up


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 video, we'll learn how to find the derivative of ln(sqrt(x)) and review the chain rule for derivatives. After we've found the derivative, we'll see how it can be applied to an everyday situation.

Steps to Solve

If we want to find the derivative of ln(√x), we will have to make use of the chain rule for derivatives. The chain rule for derivatives is a rule that we use to find the derivative of functions of the form f(g(x)).


Notice that if we let f(x) = ln(x) and g(x) = √x, then f(g(x)) = ln(√x), which tells us that we can use the chain rule to find the derivative of ln(√x).

Let's think about what else we'll need to know to find this derivative. The chain rule states that if h(x) = f(g(x)), then h ' (x) = f ' (g(x)) ⋅ g ' (x)

In our example, f(x) = ln(x) and g(x) = √x. To use the chain rule, we're going to have to find f ' (g(x)), and g ' (x). Both of these derivatives are well-known.

  • The derivative of ln(x) is 1 / x.
  • The derivative of √x is (1/2)x(-1/2), or 1/(2√x).

These facts will be helpful in our quest for the derivative. Since the derivative of ln(x) is 1/x, we have that f ' (x) = 1/x, so f ' (g(x)) = 1/(√x). Also, we know that g ' (x) = 1 / (2√x). So all we have to do is plug these into the chain rule formula and simplify to get our derivative.



Finding the derivative of ln(√x) wasn't so hard. Now, let's consider a scenario where being able to do so might come in handy. Suppose a girl named Mandi has gone through a growth spurt over the past few years. We can model her growth over these 36 months by using the function h(x) = ln(√x) + 1, where h(x) is the number of inches that she's grown during this growth spurt, and x is the number of months since the growth spurt started.

Mandi notices that when her growth spurt first began, she grew very quickly, but it seems to have slowed, though she's still growing. We can illustrate this trend by graphing the function.


This gets Mandi to thinking about just how quickly she was growing at different times over the past 36 months. We can use the derivative of ln(√x) to answer her question.

To unlock this lesson you must be a 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

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
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? 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