Math Problem Statement

What is the derivative of absolute value, y = |x|

Solution

The derivative of the absolute value function y=xy = |x| is a piecewise function because the derivative depends on whether xx is positive, negative, or zero.

Step 1: Break it into two cases:

  1. For x>0x > 0: x=x|x| = x
    • The derivative is: ddxx=1\frac{d}{dx}x = 1
  2. For x<0x < 0: x=x|x| = -x
    • The derivative is: ddx(x)=1\frac{d}{dx}(-x) = -1

Step 2: Consider the case at x=0x = 0:

The derivative at x=0x = 0 does not exist because the left-hand limit and right-hand limit do not agree:

  • As x0x \to 0^-, the derivative approaches 1-1.
  • As x0+x \to 0^+, the derivative approaches 11.

Since these two values do not match, the derivative at x=0x = 0 is undefined.

Conclusion:

The derivative of y=xy = |x| 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.

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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