Do you know the formula for the derivative of natural log? And do you know why the natural logarithm is "natural"? In this page we'll see the intuition behind this formula and learn how to solve derivatives using it.

Do you have a question or doubt about this topic? An "impossible problem"? Submit it here!

The derivative of the logarithm with base "e" is a fascinating subject in calculus. Look at this table, it might spark your curiosity:

Ok, let's get to work. You can first watch this video or read below:

We'll simply use the definition of the derivative:

Here we can apply the following property of logarithms:

Applied to our derivative:

We can also write this as:

Now we'll use this property of logs:

That is:

Now we'll use a clever substitution:

This is equivalent to:

You'll see in a moment why we use this substitution. Now, we'll note something of value. We hava a limit with Δx approaching 0. What does n approach when Δx approaches 0?

As Δx approaches 0, 1 over n approaches 0. That's equivalent to saying that n approaches infinity.

So, doing these substitutions, our derivative becomes:

We can also write this as:

Using the properties of logs again:

Now, taking the 1 over x out of the limit sign, because it doesn't depend on n:

Because the log function is continuous, we can write this as:

(If this confuses you, please review the page on continuous functions).

The limit that's inside the log function is the definition of number e:

So, the derivative simplifies to:

And the natural log of e is just one. So we're left with:

Pretty amazing result.

Let's solve some derivatives involving the natural logarithm. Let's find the derivative of:

Looks complicated? Here we'll apply the chain rule. What is the derivative of the outside function? It is one over the argument:

And the derivative of cos(x) is -sin(x):

Derivative of Natural Log, Example 2

Let's derive:

We'll apply the chain rule:

And the derivative of the inside function is just a:

Derivative of Natural Log, Example 3

Let's derive:

What would happen if you applied the chain rule directly? This would become a very long and hairy problem.

Instead, we'll use a trigonometric trick. Inside the square root, let's multiply both the numerator and denominator by 1+sin(x):

What we have in the denominator is cos squared of x. So:

This looks much easier, doesn't it? Now we can use the chain rule:

Now we'll apply the product rule in the second factor:

And now it is just algebra:

Performing the sum:

We can simplify a bunch of things:

And finally we have:

If you've found this page useful, you'll love the * Intuitive Online Calculus Course. *This is a multiple-part course that gives you the basic tools for you to master calculus.

You'll receive the first lesson immediatelly after you sign-up. A new lesson will be delivered to your e-mail inbox every second day during the duration of the course.

**Don't miss this opportunity! **This is a completelly free course in which I put my best ideas, no strings attached. As a plus, you'll get also instant access to three special resources that can be really helpful if you're serious about succeeding at calculus:

- Instant access to the special report:
**Top Ten Tips for Succeeding in Calculus**. - Instant download of
**The Intuitive Calculus Tables**: The most complete calculus tables on the web, ready for print.

Subscribe below and you'll receive instant access to the special resources and the first lesson. For more details, click here.

Conclusion

- The derivative of ln(x) is one over the argument.
- One of the reasons why the logarithm with base e is called "natural" is that its derivative is very simple.
- Do some problems involving logarithms, you might need to be familiar with the chain rule.

If you have just a general doubt about a concept, I'll try to help you. If you have a problem, or set of problems you can't solve, please send me your attempt of a solution along with your question. These will appear on a new page on the site, along with my answer, so everyone can benefit from it.

Return from **Derivative of Natural Log** to **Derivatives**

Return to **Home Page**

Sign up below to get access to the two resources only available to subscribers:

**Free Report: Top Ten Tips for Succeeding in Calculus:**everything I learned the hard way.**The Intuitive Calculus Tables:**all the formulas you'll need just in one place, ready for print.

**Plus**: *Intuitive Calculus Insights*. A free subscription to my newsletter, which I send once every two weeks. You'll receive study tips and other valuable things that'll help you succeed in calculus. **Learn more here...**

## New! Comments

Have your say about what you just read! Leave me a comment in the box below.