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AP Calculus AB/BC
Nov 23, 2025
66
3 pages
Calculus brings math to life by showing us how things change. In this unit, we'll explore applications of... Show more
![DAY 1
[e le Find the derivative of yze
sinx
sinx
•[e]=eu
Ex:yze
•COS X
ye
•x [a] -In(a).a°•. a=rational #. In(a)=constant.
• Exif(x)=x²-2² 2](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FMCIRbuIUcdctJlOHzifn_image_page_1.webp&w=2048&q=75)
Need to find derivatives quickly? These essential formulas will save you time on tests! The derivative of e^x is simply e^x itself—one of the coolest properties in calculus. When dealing with expressions like e^, use the chain rule e^ = e^4·e^x, so its derivative is e^4·e^x.
For logarithms, remember that the derivative of ln x is 1/x. When you have more complex expressions like ln(x²), use the chain rule d/dx = (2x)/(x²) = 2/x. Similarly, for logarithms with different bases, use the formula d/dx = (1)/(u ln(a))·du/dx.
Trigonometric functions follow their own patterns the derivative of sin x is cos x, and the derivative of cos x is -sin x. For exponential functions with different bases, like 4^x, the derivative is 4^x·ln(4).
Pro Tip When solving derivative problems, always identify which rule applies before starting. Most mistakes happen when students apply the wrong formula!
Basic logarithm properties are also important to remember
![DAY 1
[e le Find the derivative of yze
sinx
sinx
•[e]=eu
Ex:yze
•COS X
ye
•x [a] -In(a).a°•. a=rational #. In(a)=constant.
• Exif(x)=x²-2² 2](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FMCIRbuIUcdctJlOHzifn_image_page_2.webp&w=2048&q=75)
Ever wonder how to find where a function peaks or dips? Critical points are where f'(x) = 0 or f'(x) is undefined—they're the key to finding important features on graphs! To determine where a function increases or decreases, analyze the sign of the first derivative.
When f'(x) > 0, the function is increasing. When f'(x) < 0, the function is decreasing. For example, to analyze y = -x⁴ - 4x³ + 20x² - 10, find f'(x) = -4x³ - 12x² + 40x, then locate where this equals zero x = -5, 0, and 2. Testing points in each interval reveals where the function increases and decreases.
Concavity tells us about the shape of the graph. When f''(x) > 0, the function is concave up (smiles), and when f''(x) < 0, it's concave down (frowns). A point of inflection occurs where concavity changes—this happens when f''(x) = 0 and changes sign.
Remember When analyzing functions, create a number line with your critical points and test the sign of f'(x) in each interval—this visual approach helps avoid confusion!
For trigonometric functions like f(x) = 12 + 4x - 3tan x, the same principles apply, but you'll need to work with the appropriate domain restrictions and use trigonometric identities to solve.
![DAY 1
[e le Find the derivative of yze
sinx
sinx
•[e]=eu
Ex:yze
•COS X
ye
•x [a] -In(a).a°•. a=rational #. In(a)=constant.
• Exif(x)=x²-2² 2](/_next/image?url=https%3A%2F%2Fcontent-eu-central-1.knowunity.com%2FCONTENT%2FMCIRbuIUcdctJlOHzifn_image_page_3.webp&w=2048&q=75)
Finding where functions reach their highest or lowest points is crucial for many real-world problems! Relative extrema occur at critical points where the function changes direction. A relative maximum happens when f'(x) changes from positive to negative, and a relative minimum occurs when f'(x) changes from negative to positive.
You can confirm extrema using the second derivative test if f'(x) = 0 and f''(x) < 0, you've found a relative maximum; if f''(x) > 0, it's a relative minimum. For example, with y = x⁴ - 2x² + 8, find critical points at x = 0, ±1, then use f''(x) to classify each point.
Motion applications connect derivatives to the physical world. Position (where you are) differentiates to velocity (how fast you're moving with direction), which differentiates to acceleration (how velocity changes). Remember
Physics Connection When velocity and acceleration have the same sign, speed is increasing. When they have opposite signs, speed is decreasing. This helps analyze real-world motion problems!
Implicit differentiation helps when you can't easily solve for y. Differentiate both sides with respect to x, remembering to use the chain rule for y-terms . This technique helps find equations of tangent lines and identify horizontal and vertical tangents.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
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
Calculus brings math to life by showing us how things change. In this unit, we'll explore applications of derivatives—how to find critical points, analyze function behavior, and solve real-world problems involving rates of change.
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Need to find derivatives quickly? These essential formulas will save you time on tests! The derivative of e^x is simply e^x itself—one of the coolest properties in calculus. When dealing with expressions like e^, use the chain rule: e^ = e^4·e^x, so its derivative is e^4·e^x.
For logarithms, remember that the derivative of ln x is 1/x. When you have more complex expressions like ln(x²), use the chain rule: d/dx = (2x)/(x²) = 2/x. Similarly, for logarithms with different bases, use the formula d/dx = (1)/(u ln(a))·du/dx.
Trigonometric functions follow their own patterns: the derivative of sin x is cos x, and the derivative of cos x is -sin x. For exponential functions with different bases, like 4^x, the derivative is 4^x·ln(4).
Pro Tip: When solving derivative problems, always identify which rule applies before starting. Most mistakes happen when students apply the wrong formula!
Basic logarithm properties are also important to remember:
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Ever wonder how to find where a function peaks or dips? Critical points are where f'(x) = 0 or f'(x) is undefined—they're the key to finding important features on graphs! To determine where a function increases or decreases, analyze the sign of the first derivative.
When f'(x) > 0, the function is increasing. When f'(x) < 0, the function is decreasing. For example, to analyze y = -x⁴ - 4x³ + 20x² - 10, find f'(x) = -4x³ - 12x² + 40x, then locate where this equals zero: x = -5, 0, and 2. Testing points in each interval reveals where the function increases and decreases.
Concavity tells us about the shape of the graph. When f''(x) > 0, the function is concave up (smiles), and when f''(x) < 0, it's concave down (frowns). A point of inflection occurs where concavity changes—this happens when f''(x) = 0 and changes sign.
Remember: When analyzing functions, create a number line with your critical points and test the sign of f'(x) in each interval—this visual approach helps avoid confusion!
For trigonometric functions like f(x) = 12 + 4x - 3tan x, the same principles apply, but you'll need to work with the appropriate domain restrictions and use trigonometric identities to solve.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Finding where functions reach their highest or lowest points is crucial for many real-world problems! Relative extrema occur at critical points where the function changes direction. A relative maximum happens when f'(x) changes from positive to negative, and a relative minimum occurs when f'(x) changes from negative to positive.
You can confirm extrema using the second derivative test: if f'(x) = 0 and f''(x) < 0, you've found a relative maximum; if f''(x) > 0, it's a relative minimum. For example, with y = x⁴ - 2x² + 8, find critical points at x = 0, ±1, then use f''(x) to classify each point.
Motion applications connect derivatives to the physical world. Position (where you are) differentiates to velocity (how fast you're moving with direction), which differentiates to acceleration (how velocity changes). Remember:
Physics Connection: When velocity and acceleration have the same sign, speed is increasing. When they have opposite signs, speed is decreasing. This helps analyze real-world motion problems!
Implicit differentiation helps when you can't easily solve for y. Differentiate both sides with respect to x, remembering to use the chain rule for y-terms . This technique helps find equations of tangent lines and identify horizontal and vertical tangents.
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.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
2
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
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
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
