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Derivative of ln(x)

3 min readdecember 10, 2021

Derivative of ln(x)

This guide will show you the derivative of ln(x) and how to use this rule to help you solve even more complex derivatives! Of course, we assume (or recommend) that you understand the basic concepts of a derivative first. 😁

Formula

The formula to finding the derivative of a natural log is actually quite simple:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vH38YUMM2SwL.png?alt=media&token=2843c44c-8d5d-42a0-8232-104bac2417fc

💡 Note: This can also be written as (ln(x))’ = 1/x. Both notations mean the same thing!

Domain

We know that ln(x) is defined for all x > 0, and therefore the domain of the function is (0,∞) in interval notation. 

Well, guess what? The domain of the derivative is the same as for the original function! It makes your life just a little easier, right? The domain of 1/x is also (0,∞). 😌

Chain Rule & ln(u)

So, the derivative rule for natural log functions seems easy enough. But, what would you do instead if you had to take the natural log of an entire function? 😲

The trick here is thinking of it as a chain rule problem instead: make ln(u) have the outer function be the “ln( )” part, and the inner function be whatever is inside the parentheses.

You may recall, the way to take a chain rule derivative is:

 f[g(x)] = f’[g(x)] * g’(x)

In this case, f(x) is the natural log and g(x) is the inner function inside the parentheses.

You take the derivative of the natural log function first, which is 1/u ('u' being the original inner function), and then multiply it by the inner function's derivative. 

When we simplify this further, we get the rule for taking the derivative of the natural log of a function ln(u) = u'/u.

Practice Problems

Simpler than you thought? Let’s move on to some practice problems now! Hopefully it stays that way. 😉

1. f(x) ln(3x5)

This is a simple example of when to use the ln(u) rule instead: in this case, applying the form ln(u) = u'u gives us  (3x+5)'/(3x+5.) When you take the derivative of the top and simplify, this gives you f’(x) =3/(3x+5) for your final answer.

2. f(x) ln(x2-4)

Here’s something a little more complex! Let’s start by rewriting f(x) first as ln([x^2-4]^1/2), which creates a 2nd inner function (x^2-4) inside the first one.

However, there is an easy way to simplify this using log properties: you may recall the power rule, which states that ln(ab) = b*ln(a). Using this rule, we can rewrite f(x) once again in a simplified form as 1/2ln(x^2-4), which does look much more manageable.

Then, by using our derivative rule we get (1/2)((x^2-4)'/(x2-4)), giving us 2x/2(x^2-4). Once simplified further, we get our final answer of f’(x) = x/(x^2-4).

3. f(x) ln((6x29)/(3x3-2))

Now, I’m sure I’ve said “chain rule” too many times during this guide, but once again, chain rule! However, don’t tackle this question like a regular chain rule problem, because using quotient rule might take you years to fully solve. Try log properties instead!

Recall the quotient rule for logarithms, where ln(xy)= ln(x) - ln(y). Looks a lot simpler to solve, doesn’t it? Let’s apply it to this problem!

So, now we have f(x) = ln(6x^2+9) - ln(3x^3-2). Now, we can use chain rule to solve both natural logs, which would give us ((6x^2+9)'/(6x^2+9)) -((3x^3-2)'/(3x^3-2)). Simplifying further, we get 12x/(6x^2+9) - (9x^2/(3x^3-2)), and you can either leave f’(x) at this or you can subtract the polynomials to get a *final* final answer of f’(x) = (12x^3-8x-18x^4+27x^2)/(6x^4+5x^2-6).

Common Mistakes

Even though derivatives might look easy at first glance, they slowly become more menacing as you are presented with more complex functions to derive! Below are some common mistakes students make that you should be sure to avoid. ❗

log(x) vs. ln(x)

We know that ln(x) is the notation for a natural logarithmic function; “ln” is just another way to write “logarithm with base e.” 

In other words,

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Hk8M5JpLseOA.png?alt=media&token=b521e073-e5f5-4478-a003-5a63807039c4

You might wonder, then: why aren’t the derivatives of ln(x) and log(x) the same?

This is because when unspecified, the base of log(x) is 10. This means that the derivatives of the two functions are NOT the same. 🙅‍♀️

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ylxfzpXZttrO.png?alt=media&token=6b48e99f-1c03-41b1-8c9d-a4cf7c810bb2

🤝Connect with other students studying Calculus with Hours!

Derivative of ln(x)

3 min readdecember 10, 2021

Derivative of ln(x)

This guide will show you the derivative of ln(x) and how to use this rule to help you solve even more complex derivatives! Of course, we assume (or recommend) that you understand the basic concepts of a derivative first. 😁

Formula

The formula to finding the derivative of a natural log is actually quite simple:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vH38YUMM2SwL.png?alt=media&token=2843c44c-8d5d-42a0-8232-104bac2417fc

💡 Note: This can also be written as (ln(x))’ = 1/x. Both notations mean the same thing!

Domain

We know that ln(x) is defined for all x > 0, and therefore the domain of the function is (0,∞) in interval notation. 

Well, guess what? The domain of the derivative is the same as for the original function! It makes your life just a little easier, right? The domain of 1/x is also (0,∞). 😌

Chain Rule & ln(u)

So, the derivative rule for natural log functions seems easy enough. But, what would you do instead if you had to take the natural log of an entire function? 😲

The trick here is thinking of it as a chain rule problem instead: make ln(u) have the outer function be the “ln( )” part, and the inner function be whatever is inside the parentheses.

You may recall, the way to take a chain rule derivative is:

 f[g(x)] = f’[g(x)] * g’(x)

In this case, f(x) is the natural log and g(x) is the inner function inside the parentheses.

You take the derivative of the natural log function first, which is 1/u ('u' being the original inner function), and then multiply it by the inner function's derivative. 

When we simplify this further, we get the rule for taking the derivative of the natural log of a function ln(u) = u'/u.

Practice Problems

Simpler than you thought? Let’s move on to some practice problems now! Hopefully it stays that way. 😉

1. f(x) ln(3x5)

This is a simple example of when to use the ln(u) rule instead: in this case, applying the form ln(u) = u'u gives us  (3x+5)'/(3x+5.) When you take the derivative of the top and simplify, this gives you f’(x) =3/(3x+5) for your final answer.

2. f(x) ln(x2-4)

Here’s something a little more complex! Let’s start by rewriting f(x) first as ln([x^2-4]^1/2), which creates a 2nd inner function (x^2-4) inside the first one.

However, there is an easy way to simplify this using log properties: you may recall the power rule, which states that ln(ab) = b*ln(a). Using this rule, we can rewrite f(x) once again in a simplified form as 1/2ln(x^2-4), which does look much more manageable.

Then, by using our derivative rule we get (1/2)((x^2-4)'/(x2-4)), giving us 2x/2(x^2-4). Once simplified further, we get our final answer of f’(x) = x/(x^2-4).

3. f(x) ln((6x29)/(3x3-2))

Now, I’m sure I’ve said “chain rule” too many times during this guide, but once again, chain rule! However, don’t tackle this question like a regular chain rule problem, because using quotient rule might take you years to fully solve. Try log properties instead!

Recall the quotient rule for logarithms, where ln(xy)= ln(x) - ln(y). Looks a lot simpler to solve, doesn’t it? Let’s apply it to this problem!

So, now we have f(x) = ln(6x^2+9) - ln(3x^3-2). Now, we can use chain rule to solve both natural logs, which would give us ((6x^2+9)'/(6x^2+9)) -((3x^3-2)'/(3x^3-2)). Simplifying further, we get 12x/(6x^2+9) - (9x^2/(3x^3-2)), and you can either leave f’(x) at this or you can subtract the polynomials to get a *final* final answer of f’(x) = (12x^3-8x-18x^4+27x^2)/(6x^4+5x^2-6).

Common Mistakes

Even though derivatives might look easy at first glance, they slowly become more menacing as you are presented with more complex functions to derive! Below are some common mistakes students make that you should be sure to avoid. ❗

log(x) vs. ln(x)

We know that ln(x) is the notation for a natural logarithmic function; “ln” is just another way to write “logarithm with base e.” 

In other words,

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Hk8M5JpLseOA.png?alt=media&token=b521e073-e5f5-4478-a003-5a63807039c4

You might wonder, then: why aren’t the derivatives of ln(x) and log(x) the same?

This is because when unspecified, the base of log(x) is 10. This means that the derivatives of the two functions are NOT the same. 🙅‍♀️

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ylxfzpXZttrO.png?alt=media&token=6b48e99f-1c03-41b1-8c9d-a4cf7c810bb2

🤝Connect with other students studying Calculus with Hours!



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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.