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In this post, we will go deeply into the fascinating realm of the **derivative of absolute value**, solve its puzzles, and investigate its uses. This voyage promises intriguing insights, whether you’re a student seeking clarification or a maths enthusiast pursuing curiosity.

**Defining Derivative of Absolute Value**

**The derivative** of absolute value (function) is defined as the rate of change or the slope of a function at a specific point.

The **absolute value function** is defined as:

**derivative**of the

**absolute value function**can also be found piecewise. However, there’s a catch.

**derivative**of |x| is given by: \[ \frac{d}{dx} |x| = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \text{undefined} & \text{if } x = 0 \end{cases} \] This definition highlights the rate of change of the

**absolute value function**at any point \( x \), except at the origin, where it’s undefined due to the sharp change in direction.

### Example

Function, t(x) = |$x^2$ + 2x – 3|

### Solution

Use the chain rule. t'(x) = \[ \begin{cases} 2x + 2 & \text{if } x^2 + 2x – 3 > 0 \\ -(2x + 2) & \text{if } x^2 + 2x – 3 < 0 \\ \text{undefined} & \text{wherever } x^2 + 2x – 3 = 0 \end{cases} \]

**Properties ****of Derivative of Absolute Value**

**Derivatives** capture the rate of change or the slope of a function at a specific point. When dealing with the** absolute value function**, there are unique properties and behaviors we need to consider, mainly because of its piecewise nature. Let’s delve into the properties of the **derivative** of the **absolute value function**:

**Non-Differentiability at the Origin**

The **absolute value function $∣x∣$** has a sharp turn (or cusp) at the point $x=0$. This means that the function does not have a well-defined tangent at this point, making its **derivative** undefined at $x=0$.

**Linear Away from the Origin**

For all values x $=0$, the **absolute value function** is linear. This means that its rate of change or slope is constant in these regions.

**Positive Slope for Positive x-values**

For all $x>0$, the **derivative** of $∣x∣$ is 1. This indicates that the function is increasing at a constant rate in this interval.

**Negative Slope for Negative x-values**

For all $x<0$, the **derivative** of $∣x∣$ is -1. This means the function is decreasing at a constant rate in this interval.

**Piecewise Derivative**

**derivative**of the

**absolute value function**can be represented as a piecewise function:

### Symmetry

The **absolute value function** is symmetric about the y-axis. However, this symmetry translates into an asymmetry for its **derivative**. The rate of change is positive on the right of the y-axis and negative on the left.

### Continuity

While the **absolute value function** is continuous everywhere, its **derivative** is discontinuous at x = 0. This is a reflection of the fact that a function can be continuous at a point but not differentiable there.

### Higher-Order Derivatives

The second **derivative** of |x| does not exist for any real number because its first derivative is discontinuous at x = 0 and non-constant elsewhere.

When using the** absolute value function** in calculus, it’s essential to comprehend these characteristics, especially when dealing with issues that call for differentiation, slope analysis, or thinking about how the function behaves in respect to its **derivative.**

**Exercise**

### Example 1

Function, $f(x)=∣x∣$

Figure-2.

**Solution**

As discussed earlier,

### Example 2

### Solution

### Example 3

Function, \( h(x) = x^2 |x| \)

### Solution

Use the product rule. \[ h'(x) = x^2 \cdot f'(x) + 2x \cdot |x| = \begin{cases} 3x^2 & \text{if } x > 0 \\ -x^2 & \text{if } x < 0 \\ 0 & \text{if } x = 0 \end{cases} \]

### Example 4

Function, p(x) = |$x^2$ – 4|

### Solution

p'(x) = \[ \begin{cases} 2x & \text{if } x^2 > 4 \text{ (i.e., } x > 2 \text{ or } x < -2) \\ -2x & \text{if } x^2 < 4 \text{ (i.e., } -2 < x < 2) \\ \text{undefined} & \text{if } x = 2 \text{ or } x = -2 \end{cases} \]

### Example 5

Function, q(x) = |x| + |x-1|

### Solution

Differentiate each term individually. q'(x) = \[\begin{cases} 2 & \text{if } x > 1 \\ 0 & \text{if } x < 0 \\ 1 & \text{if } 0 \leq x \leq 1 \\ \text{undefined} & \text{if } x = 0 \text{ or } x = 1 \end{cases} \]

### Example 6

Function, r(x) = |2x + 1| – |x – 3|

### Solution

Differentiate each term individually. r'(x) = \[ \begin{cases} 1 & \text{if } x > 3 \\ 3 & \text{if } x < -1/2 \\ 2 & \text{if } -1/2 \leq x \leq 3 \\ \text{undefined} & \text{if } x = 3 \text{ or } x = -1/2 \end{cases} \]

### Example 7

Function, s(x) = $x^3$ |x|

### Solution

Use the product rule. s'(x) = \[ 3x^2 |x| + x^3 f'(x) = \begin{cases} 4x^3 & \text{if } x > 0 \\ 0 & \text{if } x = 0 \\ -2x^3 & \text{if } x < 0 \end{cases} \]

**Applications **

The d**erivative of absolute value function**, while a fundamental topic in calculus, finds its applications in several areas of study and real-world situations. Understanding the behavior of $∣x∣$ and its rate of change is essential in these contexts:

**Optimization Problems**- In economics, the
**absolute value**can represent the deviation of a certain quantity from an ideal or target value. For purposes of optimization, the**derivative**can be quite useful in determining how this deviation changes.

- In economics, the
**Signal Processing**- Understanding signal deviations and anomalies are aided by the
**absolute value function**and its**derivatives**, particularly when working with amplitude-based data.

- Understanding signal deviations and anomalies are aided by the
**Physics**- The
**absolute value**in physics can represent magnitudes or distances from reference points, particularly in the study of mechanics.

- The
**Mathematical Modelling**- Situations, where quantities deviate from a norm or target, can be modeled using the
**absolute value****function**. The**derivative**assists in understanding the sensitivity or rate of change of such deviations.

- Situations, where quantities deviate from a norm or target, can be modeled using the
**Machine Learning and Data Analysis**- In certain algorithms, the
**absolute**difference between predicted and actual values (known as the L1 norm) is minimized. The**derivative**is crucial when applying gradient-based methods to minimize this difference.

- In certain algorithms, the
**Finance**- In financial modeling, the
**absolute value**can represent deviations or differences between actual and projected values. The**derivative**can provide insights into the volatility or rate of change of such differences.

- In financial modeling, the
**Operations Research****Absolute values**are often used in formulating linear programming problems, especially in cases involving deviations from specific values. The**derivative**aids in the sensitivity analysis of these models.

**Geometry**- When studying geometric objects, the
**absolute value**can represent distances or magnitudes. The**derivative**gives information about the variation of these distances, especially in dynamic geometrical situations.

- When studying geometric objects, the
**Environmental Science**- In environmental studies, the
**absolute value**might denote deviations of certain parameters (like pollution levels) from acceptable or standard levels. The**derivative**aids in understanding the rate at which these deviations are occurring.

- In environmental studies, the
**Robust Statistics**- In statistics,
**absolute**deviations as opposed to squared deviations are sometimes used to measure dispersion because of their resistance to outliers.**Derivatives**of these metrics become relevant when looking for optimal solutions or understanding variations.

- In statistics,

*All images were created with GeoGebra.*