Subjects

AP Calculus AB/BC

Dec 13, 2025

2 pages

Fundamental Rules of Differentiation

Chelsea @zuncho

Calculus differentiation rules help you find the slope of a function without using complicated limits. These shortcuts make... Show more

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^ $n-1$ . 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.

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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Subjects

AP Calculus AB/BC

•

Dec 13, 2025

•

2 pages

Fundamental Rules of Differentiation

Chelsea

@zuncho

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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Basic Differentiation Rules

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^ $n-1$ . 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.

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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(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!

Remember: Horizontal tangent lines indicate where a function momentarily stops increasing or decreasing—like reaching the top of a hill before going down.