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.

Verified step by step guidance

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.
2. Step 2: Consider the derivative of f, denoted as f'. We want to determine the nature of f' (even, odd, or neither).
3. Step 3: Use the definition of the derivative: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
4. Step 4: Substitute -x into the derivative definition: f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(-x)}{h}.
5. 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.

Key Concepts

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.

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'.

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.