Calculus differentiation rules help you find the slope of a...
AP Calculus AB/BC46 views·Updated Jun 13, 2026·2 pagesFundamental Rules of Differentiation
1
of 2
Basic Differentiation Rules
Ever wonder how mathematicians quickly find slopes of complex curves? The answer lies in differentiation rules! Instead of using the limit definition every time, these shortcuts make calculus much more manageable.
The Constant Rule states that the derivative of any constant number is zero. Think about it: a constant function is just a horizontal line with no slope! The Power Rule is your go-to formula for expressions with variables raised to powers: if f(x) = x^n, then f'(x) = nx^. This works even with negative and fractional exponents.
When working with constants multiplied by functions, use the Constant Multiple Rule: if f(x) = c·g(x), then f'(x) = c·g'(x). This simply means you can pull constants outside the derivative. For example, the derivative of 3x² is 6x, because you multiply the constant (3) by the derivative of x² (which is 2x).
Pro Tip: When dealing with fractional exponents, convert them to their simplest form first. For example, √x is the same as x^(1/2), and ∛x is x^(1/3).
The Sum and Difference Rule lets you take derivatives term by term: the derivative of f(x) ± g(x) equals f'(x) ± g'(x). This means you can differentiate each part separately, then combine the results with the same operations.
2
of 2
More Differentiation Rules and Applications
The sine and cosine functions follow special differentiation rules you'll need to memorize: the derivative of sin(x) is cos(x), while the derivative of cos(x) is -sin(x). Notice how these functions are connected—they rotate into each other through differentiation!
Applying these rules together helps solve complex problems. For instance, when finding the derivative of f(x) = x³ + x² - 2, you simply apply the power rule to each term and get f'(x) = 3x² + 2x. For f(x) = 3x³ - 2x² + 1/x, the derivative becomes f'(x) = 9x² - 4x - 1/x².
Horizontal tangent lines occur when a function's derivative equals zero. These points often represent peaks, valleys, or inflection points on graphs. To find them, take the derivative of your function, set it equal to zero, and solve for x.
Remember: Horizontal tangent lines indicate where a function momentarily stops increasing or decreasing—like reaching the top of a hill before going down.
For example, the function f(x) = 3x² - 2x + 1 has a horizontal tangent line at x = 1/3 because f'(x) = 6x - 2 equals zero at that point. For trigonometric functions like f(x) = -cos(x), horizontal tangent lines occur at x = 0, π, 2π, and so on.
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AP Calculus AB/BC46 views·Updated Jun 13, 2026·2 pagesFundamental Rules of Differentiation
Calculus differentiation rules help you find the slope of a function without using complicated limits. These shortcuts make calculus much more efficient and practical. Let's explore the core rules that will make finding derivatives straightforward.
1
of 2
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Basic Differentiation Rules
Ever wonder how mathematicians quickly find slopes of complex curves? The answer lies in differentiation rules! Instead of using the limit definition every time, these shortcuts make calculus much more manageable.
The Constant Rule states that the derivative of any constant number is zero. Think about it: a constant function is just a horizontal line with no slope! The Power Rule is your go-to formula for expressions with variables raised to powers: if f(x) = x^n, then f'(x) = nx^. This works even with negative and fractional exponents.
When working with constants multiplied by functions, use the Constant Multiple Rule: if f(x) = c·g(x), then f'(x) = c·g'(x). This simply means you can pull constants outside the derivative. For example, the derivative of 3x² is 6x, because you multiply the constant (3) by the derivative of x² (which is 2x).
Pro Tip: When dealing with fractional exponents, convert them to their simplest form first. For example, √x is the same as x^(1/2), and ∛x is x^(1/3).
The Sum and Difference Rule lets you take derivatives term by term: the derivative of f(x) ± g(x) equals f'(x) ± g'(x). This means you can differentiate each part separately, then combine the results with the same operations.
2
of 2Sign up to see the content. It's free!
- Access to all documents
- Improve your grades
- Join milions of students
More Differentiation Rules and Applications
The sine and cosine functions follow special differentiation rules you'll need to memorize: the derivative of sin(x) is cos(x), while the derivative of cos(x) is -sin(x). Notice how these functions are connected—they rotate into each other through differentiation!
Applying these rules together helps solve complex problems. For instance, when finding the derivative of f(x) = x³ + x² - 2, you simply apply the power rule to each term and get f'(x) = 3x² + 2x. For f(x) = 3x³ - 2x² + 1/x, the derivative becomes f'(x) = 9x² - 4x - 1/x².
Horizontal tangent lines occur when a function's derivative equals zero. These points often represent peaks, valleys, or inflection points on graphs. To find them, take the derivative of your function, set it equal to zero, and solve for x.
Remember: Horizontal tangent lines indicate where a function momentarily stops increasing or decreasing—like reaching the top of a hill before going down.
For example, the function f(x) = 3x² - 2x + 1 has a horizontal tangent line at x = 1/3 because f'(x) = 6x - 2 equals zero at that point. For trigonometric functions like f(x) = -cos(x), horizontal tangent lines occur at x = 0, π, 2π, and so on.
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.
Most popular content in AP Calculus AB/BC
7AP Calculus AB/BCContinuity and One-Sided Limits
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AP Calculus AB/BCDerivative and Integral Review
Concise review for derivatives and integrals in calculus. Includes derivative rules (+ examples) and basic strategies for taking integrals in the first level of calculus (+ examples).
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AP Calculus AB/BCAP Calculus AB | Integration using U-Substitution (Indefinite Integrals)
This covers how to integrate using the u-substitution method for indefinite integrals.
1012
AP Calculus AB/BCAB Review Integrals, Area, and FTC
Review of the topics
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Notes on the topic
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Understanding the concept of derivative in context, including units, position, velocity, acceleration, related rates, and local linearity. L'HoSpital's Rule also explained.
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Analyze the initial social and religious encounters between Europeans, Africans, and Indigenous peoples in the colonial Americas.
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Analyze the environmental factors and technological innovations that led to the rise of early states in Mesopotamia, Egypt, and the Indus Valley.
9th3,1870
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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 SiOS 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 KlichAndroid 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.
AnnaiOS user