Subjects

Calculus 1

Integration and Accumulation of Change

Integrating Functions Using Long Division and Completing the Square

Calculus 1

Nov 23, 2025

4 pages

Mastering Derivatives of Inverse Trigonometric Functions in Calculus

Ella Nadine @nadenden

Inverse trigonometric functions are powerful tools that allow us to find angles when we know the trigonometric ratios.... Show more

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Derivatives of Inverse Trigonometric Functions

When finding derivatives of inverse trig functions, we use specific formulas that might look complicated at first, but follow a clear pattern. The three main formulas are

$\frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}}$

$\frac{d}{dx} \arctan u = \frac{\frac{du}{dx}}{1+ u^2}$

$\frac{d}{dx} \arcsec u = \frac{\frac{du}{dx}}{u\sqrt{u^2-1}}$

Let's see how to apply these with an example For $y = \arcsin 4x$ , we identify $u = 4x$ and $\frac{du}{dx} = 4$ . Plugging these into the formula, we get $\frac{dy}{dx} = \frac{4}{\sqrt{1- (4x)^2}} = \frac{4}{\sqrt{1- 16x^2}}$ .

Pro Tip Always start by clearly identifying what your "u" is and finding its derivative. This makes applying the formula much more straightforward!

Similarly for $y = \arctan 3x$ , with $u = 3x$ and $\frac{du}{dx} = 3$ , the derivative is $\frac{dy}{dx} = \frac{3}{1+9x^2}$ .

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

More Complex Inverse Trig Derivatives

When working with fractions inside inverse trig functions, apply the formula carefully. For example, with $y = \arctan \frac{x}{a}$ where $a$ is constant, we identify $u = \frac{x}{a}$ and find $\frac{du}{dx} = \frac{1}{a}$ .

Substituting into the formula $\frac{dy}{dx} = \frac{\frac{1}{a}}{1 + (\frac{x}{a})^2}$ . After simplifying, we get the elegant result $\frac{dy}{dx} = \frac{a}{a^2 + x^2}$ .

For problems involving products and compositions, like $x = (\arctan t)^2$ , use the chain rule. Since this is in the form $u^n$ where $u = \arctan t$ and $n = 2$ , the derivative is $\frac{dx}{dt} = \frac{2 \arctan t}{1+t^2}$ .

Remember When solving problems with products like $y = x^2 \arcsin x$ , use the product rule find the derivative of each part and combine them properly.

Products use the formula $(uv)' = u'v + uv'$ , giving us $\frac{dy}{dx} = \frac{x^2}{\sqrt{1-x^2}} + 2x \arcsin x$ for the example above.

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Complex Examples with Multiple Rules

More challenging problems combine multiple derivative rules. For example, with $y=(x-1) \sqrt{2x-x²} + \arcsin(x - 1)$ , we need both the product rule and the inverse trig formula.

For the first term, we use the product rule where $u=(x-1)$ and $v=\sqrt{2x-x²}$ . We find $u'=1$ and calculate $v'$ using the chain rule.

The second term uses the inverse sine formula with $u=x-1$ , giving us $\frac{1}{\sqrt{1-(x-1)^2}}$ .

Strategy Alert Break complex expressions into manageable pieces. Solve each piece separately using the appropriate rules, then combine your results.

After careful algebraic manipulation and combining terms, the final derivative simplifies to $\frac{2x(2-x)}{\sqrt{2x-x²}}$ , which is much cleaner than the original expression. This demonstrates how derivatives can sometimes be surprisingly simplified.

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Final Examples and Practice

The last example showcases how derivatives can help us discover interesting relationships. When finding the derivative of $y = \frac{x}{\sqrt{a^2-x^2}} - \arcsin \frac{x}{a}$ , we need both the quotient rule and the inverse trig formula.

For the first term, use the quotient rule with $u=x$ and $v=\sqrt{a^2-x^2}$ . Then calculate the derivative of $\arcsin \frac{x}{a}$ using our formula with $u=\frac{x}{a}$ .

When the terms are combined and simplified, we get the elegant result $\frac{dy}{dx} = \frac{x^2}{(a^2-x^2)^{3/2}}$ , which is surprisingly compact compared to the original expression.

You've got this! As you practice more problems, you'll develop an intuition for which techniques to apply and how terms might simplify.

The pattern of these problems shows that even complicated expressions can lead to relatively simple derivatives if you apply the rules carefully and look for opportunities to simplify.

We thought you’d never ask...

What is the Knowunity AI companion?

Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam

Quiz

Flashcards

Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.

Stefan S

iOS user

This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

Samantha Klich

Android user

Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

Anna

iOS user

I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️

Thomas R

iOS user

Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Subjects

Calculus 1

Integration and Accumulation of Change

Integrating Functions Using Long Division and Completing the Square

Calculus 1

•

Nov 23, 2025

•

4 pages

Mastering Derivatives of Inverse Trigonometric Functions in Calculus

Ella Nadine

@nadenden

Inverse trigonometric functions are powerful tools that allow us to find angles when we know the trigonometric ratios. Learning how to find the derivatives of these functions opens up ways to solve problems involving rates of change in various applications,... Show more

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Derivatives of Inverse Trigonometric Functions

When finding derivatives of inverse trig functions, we use specific formulas that might look complicated at first, but follow a clear pattern. The three main formulas are:

$\frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}}$

$\frac{d}{dx} \arctan u = \frac{\frac{du}{dx}}{1+ u^2}$

$\frac{d}{dx} \arcsec u = \frac{\frac{du}{dx}}{u\sqrt{u^2-1}}$

Let's see how to apply these with an example: For $y = \arcsin 4x$ , we identify $u = 4x$ and $\frac{du}{dx} = 4$ . Plugging these into the formula, we get $\frac{dy}{dx} = \frac{4}{\sqrt{1- (4x)^2}} = \frac{4}{\sqrt{1- 16x^2}}$ .

Pro Tip: Always start by clearly identifying what your "u" is and finding its derivative. This makes applying the formula much more straightforward!

Similarly for $y = \arctan 3x$ , with $u = 3x$ and $\frac{du}{dx} = 3$ , the derivative is $\frac{dy}{dx} = \frac{3}{1+9x^2}$ .

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

More Complex Inverse Trig Derivatives

Substituting into the formula: $\frac{dy}{dx} = \frac{\frac{1}{a}}{1 + (\frac{x}{a})^2}$ . After simplifying, we get the elegant result: $\frac{dy}{dx} = \frac{a}{a^2 + x^2}$ .

Remember: When solving problems with products like $y = x^2 \arcsin x$ , use the product rule: find the derivative of each part and combine them properly.

Products use the formula $(uv)' = u'v + uv'$ , giving us $\frac{dy}{dx} = \frac{x^2}{\sqrt{1-x^2}} + 2x \arcsin x$ for the example above.

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Complex Examples with Multiple Rules

More challenging problems combine multiple derivative rules. For example, with $y=(x-1) \sqrt{2x-x²} + \arcsin(x - 1)$ , we need both the product rule and the inverse trig formula.

For the first term, we use the product rule where $u=(x-1)$ and $v=\sqrt{2x-x²}$ . We find $u'=1$ and calculate $v'$ using the chain rule.

The second term uses the inverse sine formula with $u=x-1$ , giving us $\frac{1}{\sqrt{1-(x-1)^2}}$ .

Strategy Alert: Break complex expressions into manageable pieces. Solve each piece separately using the appropriate rules, then combine your results.

$DERIVATIVE OF INVERSE TRIGONOMETRIC FUNCTIONS $ \frac{d}{dx} \arcsin u = \frac{\frac{du}{dx}}{\sqrt{1-u^2}} $ $ \frac{d}{dx} \arctan u = \fr$

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Final Examples and Practice

For the first term, use the quotient rule with $u=x$ and $v=\sqrt{a^2-x^2}$ . Then calculate the derivative of $\arcsin \frac{x}{a}$ using our formula with $u=\frac{x}{a}$ .

When the terms are combined and simplified, we get the elegant result: $\frac{dy}{dx} = \frac{x^2}{(a^2-x^2)^{3/2}}$ , which is surprisingly compact compared to the original expression.

You've got this! As you practice more problems, you'll develop an intuition for which techniques to apply and how terms might simplify.

The pattern of these problems shows that even complicated expressions can lead to relatively simple derivatives if you apply the rules carefully and look for opportunities to simplify.

We thought you’d never ask...

What is the Knowunity AI companion?

Where can I download the Knowunity app?

You can download the app in the Google Play Store and in the Apple App Store.

Is Knowunity really free of charge?

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Smart Tools NEW

Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines

Mock Exam

Quiz

Flashcards

Essay

Can't find what you're looking for? Explore other subjects.

Students love us — and so will you.

4.9/5

App Store

4.8/5

Google Play

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user

Stefan S

iOS user

Samantha Klich

Android user

Anna

iOS user

Thomas R

iOS user

Brad T

Android user

Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍

David K

iOS user

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

Aubrey

iOS user

Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀

Marco B

iOS user

Elisha

iOS user

Paul T

iOS user