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Questions and Answers
What is the derivative of $y=5x^2$?
What is the derivative of $y=5x^2$?
5 times derivative of $x^2$
What is the derivative of a regular number?
What is the derivative of a regular number?
ZERO
What is the derivative of $(x^2 + 3)^6$ using the chain rule?
What is the derivative of $(x^2 + 3)^6$ using the chain rule?
2x times 6 $(x^2+3)^5$
What is the derivative product rule?
What is the derivative product rule?
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What is the derivative of $x^{-6}$?
What is the derivative of $x^{-6}$?
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What is the derivative of a quotient $(x^2-3)^3 / (x^{10}-10x^2)^5$?
What is the derivative of a quotient $(x^2-3)^3 / (x^{10}-10x^2)^5$?
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What is the derivative of $e^x$?
What is the derivative of $e^x$?
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What is the derivative of $ ext{ln}(x)$?
What is the derivative of $ ext{ln}(x)$?
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What is the derivative of $x^{-1}$?
What is the derivative of $x^{-1}$?
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What is the derivative of $ ext{sin}(x)$?
What is the derivative of $ ext{sin}(x)$?
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What is the derivative of $ ext{cos}(x)$?
What is the derivative of $ ext{cos}(x)$?
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What is the derivative of $ ext{tan}(x)$?
What is the derivative of $ ext{tan}(x)$?
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What is $x^{-1}$?
What is $x^{-1}$?
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How is $ ext{cos}^5(x)$ expressed?
How is $ ext{cos}^5(x)$ expressed?
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What is the formula to find the slope-intercept form of a derivative?
What is the formula to find the slope-intercept form of a derivative?
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What is the alternative formula for differentiation of $|x-1|$?
What is the alternative formula for differentiation of $|x-1|$?
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What is $5/(2x)^3$ simplified?
What is $5/(2x)^3$ simplified?
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What is the derivative of $7/(3x)^{-2}$?
What is the derivative of $7/(3x)^{-2}$?
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What is the quotient rule?
What is the quotient rule?
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What do you get when you rewrite $ rac{1/x}{x+5}$ by multiplying everything by x?
What do you get when you rewrite $ rac{1/x}{x+5}$ by multiplying everything by x?
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What is the derivative of $8(x^2 + 6)$?
What is the derivative of $8(x^2 + 6)$?
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How do you find the differential of anything?
How do you find the differential of anything?
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What does approximating function values mean?
What does approximating function values mean?
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Study Notes
Derivatives Overview
- Derivative of ( y = 5x^2 ) is calculated as ( 5 \times \frac{d}{dx}(x^2) ).
- The derivative of a constant (regular number) is always zero.
Chain Rule
- For the function ( (x^2 + 3)^6 ), the derivative is computed using:
- ( 2x \times 6(x^2 + 3)^5 ).
Product and Quotient Rules
- Product Rule:
- The derivative of a product of two functions ( f(x) ) and ( g(x) ) is ( f'(x)g(x) + f(x)g'(x) ).
- Quotient Rule applies to functions like ( \frac{(x^2-3)^3}{(x^{10}-10x^2)^5} ) and is represented as:
- ( \frac{u'v - uv'}{v^2} ), where ( u ) and ( v ) are the numerator and denominator functions, respectively.
Common Derivatives
- Derivative of ( e^x ) is ( e^x ).
- Derivative of ( \ln x ) is ( \frac{1}{x} ).
- Derivative of ( x^{-1} ) is ( -1/x^2 ).
- Derivative of ( \sin x ) is ( \cos x ).
- Derivative of ( \cos x ) is ( -\sin x ).
- Derivative of ( \tan x ) is ( \frac{1}{\cos^2 x} ) (also known as ( \sec^2 x )).
Function Transformations
- ( x^{-1} ) can be equated to ( \frac{1}{x} ).
- ( \cos^5 x ) can be expressed as ( (\cos x)^5 ).
Slope-Intercept Form
- The formula for finding the slope-intercept form is ( y - y_1 = m(x - x_1) ).
Limits and Differentiation
- For differentiation of absolute values like ( |x - 1| ):
- ( f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c} ).
Simplifying Expressions
- Expression ( \frac{5}{(2x)^3} ) simplifies to ( \frac{5}{8} x^{-3} ).
- For ( \frac{7}{(3x)^{-2}} ), the derivative evaluates to ( 126x ).
Rewriting Complex Functions
- A complex function like ( \frac{1}{x}/(x + 5) ) can be rewritten by multiplying through by ( x ):
- Results in ( \frac{x}{x^2 + 5x} ), facilitating easier differentiation.
Differential Concepts
- Find differentials by calculating derivatives: ( f' = \frac{dy}{dx} ).
- Actual differential ( dy ) is expressed as ( dy = dx \times f' ).
Approximating Function Values
- Approximating functions involves finding values like ( \sqrt{26} ) or ( 23^2 ) within the context of their formulas.
- For instance, ( \sqrt{26} ) can be approximated using ( \sqrt{25} + ) differential.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of derivatives with these flashcards from Calculus 1 Chapter 2. Each card includes a term related to differentiation and its definition, helping you reinforce your calculus knowledge. Perfect for quick revision and study sessions.
