Question

Consider \(|x|\) as a generalized function and find its derivative.

Short answer

The derivative of \(|x|\) is a piecewise function, which is \(1\) for \(x > 0\), \(-1\) for \(x < 0\), and does not exist for \(x = 0\).

Step-by-step solution

Definition of Absolute Value Function

The absolute value function, \(|x|\), is a piecewise function defined as follows: \(|x| = x\) for \(x > 0\), \(|x| = -x\) for \(x < 0\), and \(|x| = 0\) for \(x = 0\). Based on this definition, we can describe \(|x|\) as a piecewise function.

Differentiation of the Piecewise Function

The derivative of \(|x|\) will also be a piecewise function. To find this, we take derivative of each piece separately. For \(x > 0\), the derivative of \(x\) is \(1\). For \(x < 0\), the derivative of \(-x\) is \(-1\).

Conclusion

In accordance with standard mathematical procedures, at \(x = 0\), the derivative of the function \(|x|\), denoted as \( \frac{d}{dx} |x| \), fails to exist because \(|x|\) has a corner at \(x = 0\), so it is not differentiable at that point. Thereby, the derivative of \(|x|\) can be described as a piecewise function: \( \frac{d}{dx} |x| = 1\) for \(x > 0\), \(\frac{d}{dx} |x| = -1\) for \(x < 0\), and \(\frac{d}{dx} |x|\) does not exist for \(x = 0\).

Key concepts

Absolute Value Function

The absolute value function is a fundamental concept in mathematics, represented as \(|x|\). It's useful for expressing the magnitude of a number without reference to its sign. This function is crucial for handling real-world measurements where negative values don't make sense, like distance or speed.

The absolute value function is defined piecewise:

• \(|x| = x\) for \(x > 0\)
• \(|x| = -x\) for \(x < 0\)
• \(|x| = 0\) for \(x = 0\)

This piecewise definition arises because the value of \(|x|\) changes depending on whether \(x \) is positive, negative, or zero.

Understanding absolute functions helps in managing different scenarios rooms where the output must remain positive, such as finding distances between points on graphs or maps.

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each defined on a particular interval of the main function's domain. These sub-functions allow handling different conditions or situations within a single framework.

### Characteristics of Piecewise Functions

• Different Definitions: Each piece can have its own formula.
• Specific Ranges: Each formula applies to a specific part of the domain.
• Smoothness: May or may not be continuous or differentiable at the transition points.

For \(|x|\), it transitions at \(x = 0\) where it switches from \(x\) to \(-x\). Understanding how to break down functions into pieces allows students to analyze and solve complex mathematical problems more easily, focusing on smaller, simpler problems.

Derivative

The derivative is a measure of how a function changes as its input changes. In essence, it represents the slope of the tangent line at any point on the function’s graph.

### Basic Rules of Derivatives

• For \(x > 0\): The derivative of \(x\) is \(1\).
• For \(x < 0\): The derivative of \(-x\) is \(-1\).
• At \(x = 0\): The derivative does not exist due to the corner formed by the absolute value function.

The non-existence of a derivative at certain points, like at \(x = 0\) for \(|x|\), is an important concept as it indicates places where the function is not differentiable. This often occurs in functions that have sharp turns or corners. Recognizing these points is key to understanding the behavior and characteristics of a function.