Question

Prove that the derivative of an odd function is an even function and that the derivative of an even function is an odd function.

Short answer

The derivative of an odd function is even, and the derivative of an even function is odd.

Step-by-step solution

Understanding Odd and Even Functions

A function is called odd if it satisfies the condition \( f(-x) = -f(x) \) for all \( x \). A function is called even if it satisfies the condition \( f(-x) = f(x) \) for all \( x \). The task is to show how these properties affect their derivatives.

Prove Derivative of Odd Function is Even

Let \( f(x) \) be an odd function, then \( f(-x) = -f(x) \). The derivative \( f'(x) \) at any point \( x \) is defined by \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). By applying the odd function property, we have \( f(-x) = -f(x) \), so \( f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(-x)}{h} = \lim_{h \to 0} \frac{-f(x-h) + f(x)}{h} \). Simplifying shows that \( f'(-x) = \frac{d}{dx}(-f(x)) = -f'(x) \) if \( x \to -x \) and \( h \) is sufficiently small, so \( f'(-x) = f'(x) \), therefore \( f'(x) \) is even.

Prove Derivative of Even Function is Odd

Let \( f(x) \) be an even function, then \( f(-x) = f(x) \). Similarly, start with the definition of the derivative \( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \). Using the even function property: \( f(-x) = f(x) \), so \( f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(-x)}{h} = \lim_{h \to 0} \frac{f(-x+h) - f(x)}{h} \). By making a variable substitution \( u = -h \), we have that \( f'(x) = -f'(x) \) when \( x \to -x \), hence \( f'(x) \) is odd.

Key concepts

Odd and Even Functions

In mathematics, functions can be categorized as either odd or even. This classification depends on how they behave when their input is negated.

• **Odd Functions:** These satisfy the property \( f(-x) = -f(x) \) for every value of \( x \). Graphically, this means an odd function is symmetric about the origin. Examples of odd functions include \( x^3 \) and \( \sin(x) \).
• **Even Functions:** These satisfy the condition \( f(-x) = f(x) \) for all \( x \). Graphically, even functions have symmetry about the y-axis. Examples include \( x^2 \) and \( \cos(x) \).

The properties of odd and even functions significantly influence the behavior of their derivatives. Understanding these properties is fundamental for grasping their implications in calculus.

Derivative Properties

The derivative of a function tells us about the function's rate of change. When considering odd and even functions, the derivatives reveal more intriguing properties:

- **For Odd Functions:** The derivative of an odd function is even. To prove this, let's contrive the derivative \( f'(x) \). For an odd function \( f(x) \), where \( f(-x) = -f(x) \), the derivative at \(-x\) becomes \( f'(-x) = \frac{d}{dx}(-f(x)) \). Simplifying leads to \( f'(-x) = f'(x) \), showcasing the even nature of the derivative.- **For Even Functions:** The derivative of an even function is odd. For an even function \( f(x) \), satisfying \( f(-x) = f(x) \), the derivative \( f'(-x) = \lim_{h \to 0} \frac{f(-x+h) - f(x)}{h} \). This simplifies to \( f'(-x) = -f'(x) \), indicating the derivative's oddity.Recognizing these characteristics helps in understanding how function types influence their derivatives.

Functions and Continuity

Functions can be evaluated by their properties of continuity and differentiability. A function is continuous if there are no abrupt jumps or breaks in its graph.

- **Continuity:** It's a function property where, intuitively, you can draw its graph without lifting your pencil from the paper. Mathematically, a function \( f(x) \) is continuous at a point \( a \) if \( \lim_{x \to a} f(x) = f(a) \). There's also general continuity where this applies for every point in a function's domain.- **Importance in Derivatives:** Continuity ensures that the limiting process, required for defining a function's derivative, works smoothly. If a function is not continuous at a point, it cannot have a derivative at that point.Understanding these concepts is crucial for mastering calculus, as differentiability is a nuanced topic that is deeply tied with continuity. Recognizing how they influence odd and even functions' derivative behavior enhances comprehension of more advanced topics.