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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 114b
Chapter 3, Problem 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.

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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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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Related Practice
Textbook Question

Use the definition of the derivative to evaluate the following limits.

limx→25x−25x−2\(\lim\)_{x\(\to\)2}\(\frac{5^{x}\)-25}{x-2}_{}

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Textbook Question

Water flows into a conical tank at a rate of 2 ft³/min. If the radius of the top of the tank is 4 ft and the height is 6 ft, determine how quickly the water level is rising when the water is 2 ft deep in the tank.

Textbook Question

A general proof of the Chain Rule Let f and g be differentiable functions with h(x)=f(g(x)). For a given constant a, let u=g(a) and v=g(x), and define H (v) = <1x1 matrix>

c. Show that h′(a) = lim x→a ((H(g(x))+f′(g(a)))⋅g(x)−g(a)/x−a).

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Textbook Question

A spherical balloon is inflated at a rate of 10 cm³/min. At what rate is the diameter of the balloon increasing when the balloon has a diameter of 5 cm?

Textbook Question

Let f(x) = x².

a. Show that f(x)−f(y) / x−y = f′(x+y²), for all x≠y.