Solving the Derivative of ln(x)

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Solving the Derivative of ln(sqrt x)

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

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:05 Steps to Solve
  • 2:57 Application of…
  • 4:32 Lesson Summary
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.

The derivative of ln(x) is a well-known derivative. This lesson will show us the steps involved in finding this derivative, and it will go over a real world application that involves the derivative of ln(x).

Steps to Solve

We want to find the derivative of ln(x). The derivative of ln(x) is 1/x, and is actually a well-known derivative that most put to memory. However, it's always useful to know where this formula comes from, so let's take a look at the steps to actually find this derivative.

To find the derivative of ln(x), the first thing we do is let y = ln(x). Next, we use the definition of a logarithm to write y = ln(x) in logarithmic form. The definition of logarithms states that y = log b (x) is equivalent to b y = x. Therefore, by the definition of logarithms and the fact that ln(x) is a logarithm with base e, we have that y = ln(x) is equivalent to e y = x.


Okay, just a few more steps, and we'll have our formula! The next thing we want to do is treat y as a function of x, and take the derivative of each side of the equation with respect to x. We use the chain rule on the left hand side of the equation to find the derivative. The chain rule is a rule we use to take the derivative of a composition of functions, and it has two forms.


The left hand side of the equation is e y, where y is a function of x, so if we let f(x) = e x and g(x) = y, then f(g(x)) = e y. Since the derivative of e to a variable (such as e x) is the same as the original, the derivative of f'(g(x)) is e y. Therefore, by the chain rule, the derivative of e y is e y dy/dx. On the right hand side we have the derivative of x, which is 1.


We have (e y) dy/dx = 1. Now, recall that e y = x. We're going to use this fact to plug x into our equation for e y.


This gives us the equation (x)dy/dx = 1. We're getting super close now! Are you as excited as I am? We can divide both sides of this equation by x to get dy/dx = 1/x. The last thing is to recall that y = ln(x) and plug this into our equation for y.


Ta-da! Now, we see that d/dx ln(x) = 1/x, and now we know why this formula for the derivative of ln(x) is true. So what's our solution? The derivative of ln(x) is 1 / x.

Application of Derivative of ln(x)

As we said, the derivative of ln(x) is a well-known derivative that most put to memory. This is because this derivative shows up often in real world applications. Therefore, it's very useful to know the derivative so that you don't have to go through the process of finding it every time it comes up.

For example, consider a certain plane that takes off at sea level. The plane's altitude (in feet) at x minutes can be given by the function h(x) = 2000ln(x).


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 160 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