Calculus differentiation rules help you find the slope of a...
AP Calculus AB/BC46 views·Updated Aug 17, 2026·2 pagesFundamental Rules of Differentiation
1
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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^n, then f' = nx^. This works even with negative and fractional exponents.
When working with constants multiplied by functions, use the Constant Multiple Rule: if f = c·g, then f' = c·g'. 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^, and ∛x is x^.
The Sum and Difference Rule lets you take derivatives term by term: the derivative of f ± g equals f' ± g'. 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 is cos, while the derivative of cos is -sin. 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 = x³ + x² - 2, you simply apply the power rule to each term and get f' = 3x² + 2x. For f = 3x³ - 2x² + 1/x, the derivative becomes f' = 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 = 3x² - 2x + 1 has a horizontal tangent line at x = 1/3 because f' = 6x - 2 equals zero at that point. For trigonometric functions like f = -cos, horizontal tangent lines occur at x = 0, π, 2π, and so on.
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AP Calculus AB/BC46 views·Updated Aug 17, 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.
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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^n, then f' = nx^. This works even with negative and fractional exponents.
When working with constants multiplied by functions, use the Constant Multiple Rule: if f = c·g, then f' = c·g'. 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^, and ∛x is x^.
The Sum and Difference Rule lets you take derivatives term by term: the derivative of f ± g equals f' ± g'. This means you can differentiate each part separately, then combine the results with the same operations.
2
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More Differentiation Rules and Applications
The sine and cosine functions follow special differentiation rules you'll need to memorize: the derivative of sin is cos, while the derivative of cos is -sin. 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 = x³ + x² - 2, you simply apply the power rule to each term and get f' = 3x² + 2x. For f = 3x³ - 2x² + 1/x, the derivative becomes f' = 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 = 3x² - 2x + 1 has a horizontal tangent line at x = 1/3 because f' = 6x - 2 equals zero at that point. For trigonometric functions like f = -cos, horizontal tangent lines occur at x = 0, π, 2π, and so on.
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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.
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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.
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