If y=|x|, then find dy/dx.

Answer

To find the derivative of the function \( y = |x| \), we will analyze the function based on the definition of the absolute value and differentiate it piecewise.

### Step-by-Step Solution:

1. **Understanding the Absolute Value Function**:
The absolute value function \( y = |x| \) can be expressed in piecewise form:
\[ y = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \]

2. **Differentiating for \( x > 0 \)**:
For \( x > 0 \), we have:
\[ y = x \]
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{d}{dx}(x) = 1 \]

3. **Differentiating for \( x < 0 \)**:
For \( x < 0 \), we have:
\[ y = -x \]
The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{d}{dx}(-x) = -1 \]

4. **Combining the Results**:
Therefore, we can summarize the derivative as:
\[ \frac{dy}{dx} = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \end{cases} \]
Note that at \( x = 0 \), the derivative is not defined because the function has a cusp.

### Final Answer:
\[ \frac{dy}{dx} = \begin{cases} 1 & \text{if } x > 0 \\ -1 & \text{if } x < 0 \end{cases} \]