Derivatives: Key Concepts and Rules
This page provides a comprehensive overview of derivatives in mathematics, focusing on essential rules and formulas for calculating derivatives of various functions. The content is crucial for students studying calculus and advanced mathematics.
The document begins by defining the concept of a derivative. A function f is considered derivable on an interval I of R if and only if f'a can be calculated at every point of I. This foundational definition sets the stage for understanding the subsequent rules and formulas.
Definition: A function f is derivable on an interval I if f'a can be calculated for all points in I.
The page then presents several key rules for calculating derivatives:
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Derivative of a Quotient: The formula u/v' = v⋅u′−u⋅v′ / v² is provided, which is essential for differentiating complex fractions.
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Derivative of a Sum: The rule u+v' = u' + v' is given, showing that the derivative of a sum is the sum of the derivatives.
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Derivative of a Product: The formula u⋅v' = u'·v + u·v' is presented, crucial for differentiating more complex functions.
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Derivative of the Inverse of a Function: The rule 1/u' = -u' / u² is provided, useful for functions involving reciprocals.
Highlight: These rules form the foundation for differentiating complex functions and are essential for solving advanced calculus problems.
The document also includes a section on derivatives of common functions:
- Constant function: fx = a → f'x = 0
- Linear function: fx = ax + b → f'x = a
- Quadratic function: fx = x² → f'x = 2x
- Power function: fx = xⁿ → f'x = n·xⁿ⁻¹
- Square root function: fx = √x → f'x = 1 / 2√x
- Reciprocal function: fx = 1/x → f'x = -1/x²
Example: For the function fx = x², its derivative is f'x = 2x, demonstrating the power rule.
These examples provide practical applications of the derivative rules and help students understand how to apply them to various functions.
Vocabulary: Dérivée des fonctions usuelles pdf Derivativeofcommonfunctionspdf and Formules de dérivées pdf Derivativeformulaspdf are key terms related to this content, often searched for by students looking for comprehensive resources on derivatives.
The page serves as a valuable reference for students studying calculus, providing a Tableau des dérivées usuelles Tableofcommonderivatives that can be quickly consulted during problem-solving. The inclusion of both rules and examples makes this resource particularly useful for understanding the practical application of derivative concepts.
