Comprehensive Guide to Derivative Formulas
This page presents a detailed Tableau des dérivées usuelles, offering a systematic approach to understanding and applying derivative rules in calculus. The table is organized into three columns: the original function f(x), its derivative f'(x), and the domain of differentiability.
The guide begins with simple functions and progresses to more complex ones:
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Constant Function: For f(x) = k, where k is a real number, the derivative is always 0, and it's differentiable over all real numbers.
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Identity Function: When f(x) = x, the derivative is 1, also differentiable over all real numbers.
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Linear Function: For f(x) = ax + b, the derivative is simply a, the slope of the line, differentiable over all real numbers.
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Quadratic Function: The derivative of f(x) = x² is f'(x) = 2x, differentiable over all real numbers.
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Power Function: For f(x) = xⁿ, where n is a positive integer, the derivative is f'(x) = nxⁿ⁻¹, differentiable over all real numbers.
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Square Root Function: The derivative of f(x) = √x is f'(x) = 1/(2√x), differentiable over the positive real numbers.
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Trigonometric Functions:
- For f(x) = sin x, the derivative is f'(x) = cos x
- For f(x) = cos x, the derivative is f'(x) = -sin x
Both are differentiable over all real numbers.
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Reciprocal Function: For f(x) = 1/x, the derivative is f'(x) = -1/x², differentiable over all non-zero real numbers.
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Negative Power Function: For f(x) = x⁻ⁿ, the derivative is f'(x) = -nx⁻ⁿ⁻¹, differentiable over all non-zero real numbers.
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Composite Function: For f(x) = v(u(x)), the derivative is f'(x) = u'(x) × v'(u(x)), following the chain rule. This is differentiable when u is differentiable on an interval I with values in J, and v is differentiable on J.
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Exponential Function: For f(x) = eˣ, the derivative is f'(x) = eˣ, differentiable over all real numbers.
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Natural Logarithm: For f(x) = ln x, the derivative is f'(x) = 1/x, differentiable over all positive real numbers.
Highlight: The Tableau dérivée Terminale provides a comprehensive overview of derivative rules, essential for advanced calculus and mathematical analysis.
Example: For the quadratic function f(x) = x², the derivative f'(x) = 2x can be used to find the slope of the tangent line at any point on the parabola.
Vocabulary: The "domain of differentiability" refers to the set of x-values for which the derivative of a function exists and is continuous.
Definition: The chain rule, used for composite functions, states that the derivative of a composite function is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function.
This Tableau des dérivées et primitives PDF serves as a valuable resource for students and professionals alike, offering a quick reference for common derivative formulas and their domains of applicability.
