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elisa
@elisa_xghl
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The document provides a comprehensive overview of derivative formulas and fundamental concepts in calculus. It covers the definition of derivatives, their geometric interpretation, and various rules for calculating derivatives of different functions. The material is presented with detailed explanations, examples, and proofs, making it an excellent resource for students studying analisi matematica.
1/1/2023
15580
This page provides detailed proofs for some of the differentiation rules presented earlier. The proofs use the definition of the derivative and algebraic manipulation.
Key proofs include:
Example: The proof of the product rule uses the limit definition and clever algebraic rearrangement to arrive at the final formula.
The page also includes examples of applying these rules to specific functions, such as y = 8cos x and y = 3x⁴.
Highlight: The product rule proof introduces a technique of adding and subtracting the same term to facilitate the limit calculation, which is a common strategy in more advanced proofs.
This page provides detailed proofs for some of the derivate fondamentali presented earlier. The proofs use the definition of the derivative as the limit of the incremental ratio.
Key proofs include:
Example: For y = x², the proof shows that y' = 2x using the limit definition and algebraic manipulation.
The page also includes examples of applying these proofs to specific functions, such as y = x² and y = x³.
Highlight: The general formula for the derivative of xⁿ is derived as y' = n·xⁿ⁻¹, which is a fundamental result in calculus.
This page continues with proofs of differentiation rules, focusing on the quotient rule and the chain rule.
Key proofs include:
Highlight: The proof of the chain rule uses a change of variable technique, introducing z = g(x), which simplifies the limit calculation.
The page emphasizes the importance of these rules in calculating derivatives of complex functions. It also includes examples of applying the quotient rule to specific functions.
Example: The derivative of y = 5/e^x is calculated using the quotient rule, resulting in y' = -5e^(-x).
This page continues with proofs of fundamental derivatives, focusing on more complex functions.
Key proofs include:
Highlight: The proof for the derivative of sin x uses important trigonometric identities and limit theorems.
The page emphasizes the use of limit theorems and algebraic manipulation to derive these important formulas. It also introduces the concept of notable limits, which are crucial for many derivative calculations.
Vocabulary: Notable limits are specific limit expressions that frequently appear in calculus and have known values, such as lim[x→0] (sin x) / x = 1.
This page introduces the concept of derivatives and their geometric interpretation.
The derivative of a function at a point is defined as the limit of the rapporto incrementale (incremental ratio) as the increment approaches zero. Geometrically, the derivative represents the slope of the tangent line to the function's graph at a given point.
Definition: The derivative of a function f(x) at x₀ is defined as: f'(x₀) = lim[h→0] [f(x₀+h) - f(x₀)] / h
Key concepts covered include:
Highlight: A function is differentiable on a closed interval [a,b] if it is differentiable at all interior points and has finite left and right derivatives at the endpoints.
The page also includes visual representations of the secant line approaching the tangent line as the increment h approaches zero.
This final page provides practice problems for calculating derivatives, including one-sided derivatives and derivatives of more complex functions. It includes step-by-step solutions to guide students through the problem-solving process.
Key types of problems:
Example: One problem asks to calculate the left-hand and right-hand derivatives of f(x) = |-x² + 2x| at x = 2.
The page emphasizes the importance of carefully analyzing the function's behavior and choosing the appropriate differentiation techniques for each problem.
Highlight: When dealing with functions involving absolute values, it's essential to consider the function's definition on either side of the point where the absolute value becomes zero.
This page presents a table of derivate fondamentali (fundamental derivatives) for common functions. These formulas are essential for calculating derivatives of more complex functions.
Key derivatives covered include:
Example: The derivative of y = √x is y' = 1 / (2√x)
The page also includes derivatives of inverse trigonometric functions, which are important for solving more advanced problems in calculus.
This page presents the rules for differentiating combinations of functions, known as derivate composte (composite derivatives). These rules are essential for calculating derivatives of more complex functions.
Key rules covered include:
Example: The derivative of y = e^(f(x)) is y' = e^(f(x)) · f'(x)
The page also includes formulas for derivatives of composite functions involving logarithms, trigonometric functions, and inverse trigonometric functions.
This page focuses on calculating one-sided derivatives for piecewise-defined functions. It provides examples and step-by-step solutions for finding left-hand and right-hand derivatives at specific points.
Key concepts covered:
Example: For the function f(x) = {x-3 if x≤3, x-1 if x>3}, the left-hand and right-hand derivatives at x=3 are calculated separately.
The page also includes an example of finding derivatives for a function involving absolute values, which requires careful consideration of the function's behavior on either side of the point where the absolute value changes.
Highlight: When dealing with piecewise functions or functions involving absolute values, it's crucial to consider the behavior of the function on both sides of the point of interest.
126
2931
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Integrali
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5363
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derivate
schemi derivate
270
7207
5ªl
Studio di funzione
studio di una funzione fratta Dominio Segno intersezione assi limiti asintoti derivata prima e seconda
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4ªl/5ªl
Derivate
:)
145
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Derivata
Derivata: evoluzione storica (Newton e Libeniz), il concetto di derivata, derivate elementari, operazioni con le derivate, derivabilità, punti di non derivabilità, introduzione al differenziale
Valutazione media dell'app
Studenti che usano Knowunity
Nelle classifiche delle app per l'istruzione in 12 Paesi
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elisa
@elisa_xghl
·
363 Follower
Segui
The document provides a comprehensive overview of derivative formulas and fundamental concepts in calculus. It covers the definition of derivatives, their geometric interpretation, and various rules for calculating derivatives of different functions. The material is presented with detailed explanations, examples, and proofs, making it an excellent resource for students studying analisi matematica.
This page provides detailed proofs for some of the differentiation rules presented earlier. The proofs use the definition of the derivative and algebraic manipulation.
Key proofs include:
Example: The proof of the product rule uses the limit definition and clever algebraic rearrangement to arrive at the final formula.
The page also includes examples of applying these rules to specific functions, such as y = 8cos x and y = 3x⁴.
Highlight: The product rule proof introduces a technique of adding and subtracting the same term to facilitate the limit calculation, which is a common strategy in more advanced proofs.
This page provides detailed proofs for some of the derivate fondamentali presented earlier. The proofs use the definition of the derivative as the limit of the incremental ratio.
Key proofs include:
Example: For y = x², the proof shows that y' = 2x using the limit definition and algebraic manipulation.
The page also includes examples of applying these proofs to specific functions, such as y = x² and y = x³.
Highlight: The general formula for the derivative of xⁿ is derived as y' = n·xⁿ⁻¹, which is a fundamental result in calculus.
This page continues with proofs of differentiation rules, focusing on the quotient rule and the chain rule.
Key proofs include:
Highlight: The proof of the chain rule uses a change of variable technique, introducing z = g(x), which simplifies the limit calculation.
The page emphasizes the importance of these rules in calculating derivatives of complex functions. It also includes examples of applying the quotient rule to specific functions.
Example: The derivative of y = 5/e^x is calculated using the quotient rule, resulting in y' = -5e^(-x).
This page continues with proofs of fundamental derivatives, focusing on more complex functions.
Key proofs include:
Highlight: The proof for the derivative of sin x uses important trigonometric identities and limit theorems.
The page emphasizes the use of limit theorems and algebraic manipulation to derive these important formulas. It also introduces the concept of notable limits, which are crucial for many derivative calculations.
Vocabulary: Notable limits are specific limit expressions that frequently appear in calculus and have known values, such as lim[x→0] (sin x) / x = 1.
This page introduces the concept of derivatives and their geometric interpretation.
The derivative of a function at a point is defined as the limit of the rapporto incrementale (incremental ratio) as the increment approaches zero. Geometrically, the derivative represents the slope of the tangent line to the function's graph at a given point.
Definition: The derivative of a function f(x) at x₀ is defined as: f'(x₀) = lim[h→0] [f(x₀+h) - f(x₀)] / h
Key concepts covered include:
Highlight: A function is differentiable on a closed interval [a,b] if it is differentiable at all interior points and has finite left and right derivatives at the endpoints.
The page also includes visual representations of the secant line approaching the tangent line as the increment h approaches zero.
This final page provides practice problems for calculating derivatives, including one-sided derivatives and derivatives of more complex functions. It includes step-by-step solutions to guide students through the problem-solving process.
Key types of problems:
Example: One problem asks to calculate the left-hand and right-hand derivatives of f(x) = |-x² + 2x| at x = 2.
The page emphasizes the importance of carefully analyzing the function's behavior and choosing the appropriate differentiation techniques for each problem.
Highlight: When dealing with functions involving absolute values, it's essential to consider the function's definition on either side of the point where the absolute value becomes zero.
This page presents a table of derivate fondamentali (fundamental derivatives) for common functions. These formulas are essential for calculating derivatives of more complex functions.
Key derivatives covered include:
Example: The derivative of y = √x is y' = 1 / (2√x)
The page also includes derivatives of inverse trigonometric functions, which are important for solving more advanced problems in calculus.
This page presents the rules for differentiating combinations of functions, known as derivate composte (composite derivatives). These rules are essential for calculating derivatives of more complex functions.
Key rules covered include:
Example: The derivative of y = e^(f(x)) is y' = e^(f(x)) · f'(x)
The page also includes formulas for derivatives of composite functions involving logarithms, trigonometric functions, and inverse trigonometric functions.
This page focuses on calculating one-sided derivatives for piecewise-defined functions. It provides examples and step-by-step solutions for finding left-hand and right-hand derivatives at specific points.
Key concepts covered:
Example: For the function f(x) = {x-3 if x≤3, x-1 if x>3}, the left-hand and right-hand derivatives at x=3 are calculated separately.
The page also includes an example of finding derivatives for a function involving absolute values, which requires careful consideration of the function's behavior on either side of the point where the absolute value changes.
Highlight: When dealing with piecewise functions or functions involving absolute values, it's crucial to consider the behavior of the function on both sides of the point of interest.
Matematica - Integrali
schema formule integrali per ripasso
126
2931
6
Matematica - derivate
schemi derivate
142
5363
3
Matematica - Studio di funzione
studio di una funzione fratta Dominio Segno intersezione assi limiti asintoti derivata prima e seconda
270
7207
2
Matematica - Derivate
:)
7
115
1
Matematica - Derivata
Derivata: evoluzione storica (Newton e Libeniz), il concetto di derivata, derivate elementari, operazioni con le derivate, derivabilità, punti di non derivabilità, introduzione al differenziale
145
3399
0
Valutazione media dell'app
Studenti che usano Knowunity
Nelle classifiche delle app per l'istruzione in 12 Paesi
Studenti che hanno caricato appunti
Utente iOS
Stefano S, utente iOS
Susanna, utente iOS
