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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 3 - DerivativesProblema 114b
Capitolo 3, Problema 114b

Derivatives of even and odd functions Recall that f is even if f(−x) = f(x), for all x in the domain of f, and f is odd if f(−x) = −f(x) for all x in the domain of f.
b. If f is a differentiable, odd function on its domain, determine whether f' is even, odd, or neither.

Guida verificata passo dopo passo
1
Step 1: Recall the definition of an odd function. A function f is odd if f(-x) = -f(x) for all x in its domain.
Step 2: Consider the derivative of f, denoted as f'. We want to determine the nature of f' (even, odd, or neither).
Step 3: Use the definition of the derivative: f'(x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(x+h) - f(x)}{h}\).
Step 4: Substitute -x into the derivative definition: f'(-x) = \(\lim\)_{h \(\to\) 0} \(\frac{f(-x+h) - f(-x)}{h}\).
Step 5: Use the property of odd functions: f(-x+h) = -f(x-h) and f(-x) = -f(x). Substitute these into the expression for f'(-x) and simplify to determine if f'(-x) = f'(x), f'(-x) = -f'(x), or neither.

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Even and Odd Functions

A function f is classified as even if it satisfies the condition f(−x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, a function is odd if it meets the condition f(−x) = −f(x), indicating that its graph is symmetric about the origin. Understanding these definitions is crucial for analyzing the behavior of functions and their derivatives.
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Properties of Functions

Differentiability

A function is said to be differentiable at a point if it has a defined derivative at that point, which implies that the function is smooth and continuous in the vicinity of that point. Differentiability is a stronger condition than continuity; a function can be continuous but not differentiable. In the context of the question, knowing that f is differentiable allows us to explore the properties of its derivative f'.
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Finding Differentials

Properties of Derivatives of Odd Functions

The derivative of an odd function inherits certain properties from the original function. Specifically, if f is an odd function, then its derivative f' is also an odd function. This means that f' will satisfy the condition f'(-x) = -f'(x) for all x in its domain. This property is essential for determining the nature of the derivative in the context of the given problem.
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