Math Problem Statement
What is the derivative of absolute value, y = |x|
Solution
The derivative of the absolute value function is a piecewise function because the derivative depends on whether is positive, negative, or zero.
Step 1: Break it into two cases:
- For :
- The derivative is:
- For :
- The derivative is:
Step 2: Consider the case at :
The derivative at does not exist because the left-hand limit and right-hand limit do not agree:
- As , the derivative approaches .
- As , the derivative approaches .
Since these two values do not match, the derivative at is undefined.
Conclusion:
The derivative of is:
\begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \\ \text{undefined} & \text{if } x = 0 \end{cases}$$ Would you like a more detailed explanation of piecewise functions, or any related questions? Here's a set of questions that could expand your knowledge: 1. How would you calculate the derivative of a piecewise function in general? 2. What are the properties of non-differentiable points in a function? 3. How does the concept of the absolute value function apply to optimization problems? 4. Can you think of any real-world scenarios where the absolute value function would be useful? 5. How does the derivative of the absolute value function relate to the concept of slopes or rates of change? **Tip:** Always consider the behavior of a function near points where it's not continuous or differentiable, such as absolute value at 0.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Piecewise Functions
Absolute Value
Formulas
d/dx |x| = 1 for x > 0
d/dx |x| = -1 for x < 0
Theorems
Differentiability at discontinuities
Piecewise differentiable functions
Suitable Grade Level
Grades 11-12