Question

The derivative of the absolute value of x² is:

A) 2x
B) -2x
C) 2|x|
D) -2|x|

Answer 1

The derivative of the absolute value of x² is 2x.

Step-by-step explanation:

The derivative of the absolute value of x² is 2x.

To find the derivative of the absolute value function, we need to use the chain rule. Let's break it down step by step:

1. 
   
2. Start with the function f(x) = |x²|.
3. 
   
4. If x² ≥ 0, then f(x) = x².
5. 
   
6. If x² < 0, then f(x) = -x².
7. 
   
8. Take the derivative of each case separately:
9. 
   

• 
  
• If x² ≥ 0, then f'(x) = 2x.
• 
  
• If x² < 0, then f'(x) = -2x.
• 
  

Combine the derivative using the piecewise definition:

• 
  
• f'(x) = 2x for x ≥ 0.
• 
  
• f'(x) = -2x for x < 0.
•