Question

Show that: \[ \frac{d}{dx} \left(|x|\right) = \frac{x}{|x|}, \quad x \neq 0. \]

Hint:

The derivative of the absolute value function is undefined at \( x = 0 \) because \( |x| \) has a cusp at \( x = 0 \).

Answer 1

Step 1: Define the absolute value function.
The absolute value function \( |x| \) is defined as: \[ |x| = \begin{cases} x, & x>0,
-x, & x<0. \end{cases} \] Step 2: Differentiate for \( x>0 \).
For \( x>0 \), \( |x| = x \). The derivative is: \[ \frac{d}{dx} (|x|) = \frac{d}{dx} (x) = 1. \] Step 3: Differentiate for \( x<0 \).
For \( x<0 \), \( |x| = -x \). The derivative is: \[ \frac{d}{dx} (|x|) = \frac{d}{dx} (-x) = -1. \] Step 4: Combine the results.
For both cases (\( x>0 \) and \( x<0 \)), the derivative can be written as: \[ \frac{d}{dx} (|x|) = \frac{x}{|x|}. \] Step 5: Exclude \( x = 0 \).
At \( x = 0 \), the derivative is undefined because \( \frac{x}{|x|} \) involves division by zero. Therefore, this result holds only for \( x \neq 0 \). Conclusion:
The result is shown: \[ \boxed{\frac{d}{dx} (|x|) = \frac{x}{|x|}, \quad x \neq 0}. \]